Irrationality Measures to some constants of the form int( 1/(a+b*x+c*x^2),x=0..1) with 1<=a,b,c <=, 10



                              By Shalosh B. Ekhad



                             Consider the constant



                                  1
                                 /
                                |         1
                           C =  |   -------------- dx
                                |      2
                               /    c x  + b x + a
                                 0



with a,b,c, integers. We are interested in the diophantine approximations in\
    duced by the sequence



                               1
                              /         n        n
                             |         x  (1 - x)
                     E(n) =  |   ----------------------- dx
                             |       2           (n + 1)
                            /    (c x  + b x + a)
                              0



                           that can be expressed as



                                E(n)=A(n)+B(n)*C



                for sequences of rational numbers A(n) and B(n)





Using the Almkvist-Zeilberger algorithm it follows that E(n), and hence A(n)  and B(n) satisfy

          the linear recurrence equation with polynomial coefficients



                                                                 2
(-n - 1) F(n) + (2 n + 3) (2 a + b) F(n + 1) + (n + 2) (4 a c - b ) F(n + 2) =  0



                but of course with different initial conditions.





                   It follows from the Poincarre lemma that



            A(n),B(n)=OMEGA(|C1(a,b,c)|^n), E(n)=OMEGA( |C2(a,b,c)|^n )



where C1(a,b,c) is the larger root (in absolute value) and C2(a,b,c) the smaller root in abslute value) of the indicial equation of the constant-coefficient

                 recurrence approximating the above recurrence.

                 That indicial polynomial, in N,  happens to be



                                                     2   2
                     1 + (-4 a - 2 b) N + (-4 a c + b ) N



                              and in Maple format



1+(-4*a-2*b)*N+(-4*a*c+b^2)*N^2


Setting it equal to 0, and solving the quadratic equation, we get that the smaller root, C2(a,b,c) is



                                       2             1/2
                        -2 a - b + 2 (a  + a b + a c)
                       | --------------------------------|
                                            2
                                   4 a c - b



                       and the larger root, C1(a,b,c) is



                                       2             1/2
                         2 a + b + 2 (a  + a b + a c)
                     |   -------------------------------|
                                            2
                                   4 a c - b



                  and in Maple format, they are respectively,



(abs(-2*a-b+2*(a^2+a*b+a*c)^(1/2))/(4*a*c-b^2)),abs(-(2*a+b+2*(a^2+a*b+a*c)^(1/2))/(4*a*c-b^2))


Suppose that we discover that there exists a constant K (that is either an integer or a square-root of an integer) such that



     A1(n)=lcm(1..n)*K^n*A(n) and B1(n)=lcm(1..n)*K^n*B(n) are all integers.



         (if K is a square-root of an integer only do it for even n .)



                      Let E1(n)=lcm(1..n)*K^n*E(n) , then



                              E1(n)=A1(n)+B1(n)*c



                    where now A1(n) and B1(n) are INTEGERS.



                      Since lcm(1..n)=OMEGA(e^n), we have



   A1(n),B1(n)=OMEGA(K^n*e*C1(a,b,c))^n ), E(n)=OMEGA( (K^n*e*C2(a,b,c))^n )



                                     Hence

              E(n)=OMEGA(1/max(A(n),B(n))^delta ), where delta is



                    -log(K*e*C2(a,b,c))/log(K*e*C1(a,b,c))





                         |                2             1/2 |
                         | -2 a - b + 2 (a  + a b + a c)    |
                      ln(| -------------------------------- |) + ln(K) + 1
                         |                     2            |
                         |            4 a c - b             |
      that equals , - ----------------------------------------------------
                         |               2             1/2 |
                         | 2 a + b + 2 (a  + a b + a c)    |
                      ln(| ------------------------------- |) + ln(K) + 1
                         |                    2            |
                         |           4 a c - b             |



                           and in Maple format it is



-(ln(abs((-2*a-b+2*(a^2+a*b+a*c)^(1/2))/(4*a*c-b^2)))+ln(K)+1)/(ln(abs((2*a+b+2*(a^2+a*b+a*c)^(1/2))/(4*a*c-b^2)))+ln(K)+1)


                 So it is good news whenever this is positive.



By searching for quadratic polynomials P with positive coefficients <=, 10,

    we found the following list of , 89, lucky cases.



                          For each of the lists below

The first entry is the  polynomial in x, P(x), a certain irreducible quadrat ic.



the second entry is the constant whose irrationality we are claiming , namely the integral of 1/P(x) from x=0 to x=1.



The third entry is the magic constant K, mentioned above such that multiplying by K^n*lcm(1...n) makes everything integers.



                    the fourth is the exact value of delta,



 the fifth, and  last, entry is the implied irrationality measure, i.e. 1+1/delta,  in floating-point.





                                 |         1/2 |
                                 |      2 3    |        1/2
                 1/2          ln(| -1 + ------ |) + ln(3   ) + 1
  2          Pi 3      1/2       |        3    |
[x  + x + 1, -------, 3   , - ----------------------------------,
                9                |        1/2 |
                                 |     2 3    |        1/2
                              ln(| 1 + ------ |) + ln(3   ) + 1
                                 |       3    |

    8.3099863401554735233]





                                   1/2
  2                 1/2         3 5            1/2          1/2    1/2
[x  + 3 x + 1, 2/5 5    arccoth(------) - 2/5 5    arccoth(5   ), 5   ,
                                  5

         |        1/2 |
         |     2 5    |        1/2
      ln(| 1 - ------ |) + ln(5   ) + 1
         |       5    |
    - ----------------------------------, 6.5082057497631049136]
         |         1/2 |
         |      2 5    |        1/2
      ln(| -1 - ------ |) + ln(5   ) + 1
         |        5    |





                         1/2
    2              2 Pi 3                    1/2   1/2
[3 x  + 3 x + 1, - --------- + 2/3 arctan(3 3   ) 3   , 3,
                       9

         |            1/2 |
         |         2 7    |
      ln(| - 5/3 + ------ |) + ln(3) + 1
         |           3    |
    - ----------------------------------, 15.315550900340627733]
          |          1/2 |
          |       2 7    |
       ln(| 5/3 + ------ |) + ln(3) + 1
          |         3    |





                                1/2
    2                        2 3      1/2                 1/2   1/2     1/2
[7 x  + 4 x + 1, -1/3 arctan(------) 3    + 1/3 arctan(3 3   ) 3   , 2 3   ,
                               3

         |           1/2 |
         |         12    |          1/2
      ln(| - 1/2 + ----- |) + ln(2 3   ) + 1
         |           6   |
    - --------------------------------------, 8.3099863401554735302]
          |         1/2 |
          |       12    |          1/2
       ln(| 1/2 + ----- |) + ln(2 3   ) + 1
          |         6   |





    2                 1/2          1/2                   1/2   1/2
[5 x  + 5 x + 1, 2/5 5    arccoth(5   ) - 2/5 arccoth(3 5   ) 5   , 5,

          |           1/2 |
          |       2 11    |
       ln(| 7/5 - ------- |) + ln(5) + 1
          |          5    |
    - -----------------------------------, 1176.2902524845007739]
         |             1/2 |
         |         2 11    |
      ln(| - 7/5 - ------- |) + ln(5) + 1
         |            5    |





                                1/2                        1/2
    2                        5 3      1/2              19 3      1/2
[7 x  + 5 x + 1, -2/3 arctan(------) 3    + 2/3 arctan(-------) 3   , 3,
                               3                          3

         |             1/2 |
         |         2 13    |
      ln(| - 7/3 + ------- |) + ln(3) + 1
         |            3    |
    - -----------------------------------, 7.5789513504094748015]
          |           1/2 |
          |       2 13    |
       ln(| 7/3 + ------- |) + ln(3) + 1
          |          3    |





                                1/2
    2                        5 7      1/2                 1/2   1/2   1/2
[8 x  + 5 x + 1, -2/7 arctan(------) 7    + 2/7 arctan(3 7   ) 7   , 7   ,
                               7

         |          1/2 |
         |      2 14    |        1/2
      ln(| -1 + ------- |) + ln(7   ) + 1
         |         7    |
    - -----------------------------------, 4.8569705938289177946]
         |         1/2 |
         |     2 14    |        1/2
      ln(| 1 + ------- |) + ln(7   ) + 1
         |        7    |





                                                   |        1/2 |
                                                   |      17    |
                                                ln(| -2 + ----- |) + ln(4) + 1
     2                                             |        2   |
[10 x  + 6 x + 1, -arctan(3) + arctan(13), 4, - ------------------------------,
                                                   |       1/2 |
                                                   |     17    |
                                                ln(| 2 + ----- |) + ln(4) + 1
                                                   |       2   |

    10.432735096002347812]





                                1/2
    2                        3 7      1/2               1/2   1/2   1/2
[2 x  + 3 x + 2, -2/7 arctan(------) 7    + 2/7 arctan(7   ) 7   , 7   ,
                               7

         |          1/2 |
         |      2 14    |        1/2
      ln(| -1 + ------- |) + ln(7   ) + 1
         |         7    |
    - -----------------------------------, 4.8569705938289177946]
         |         1/2 |
         |     2 14    |        1/2
      ln(| 1 + ------- |) + ln(7   ) + 1
         |        7    |





                                1/2
    2                        3 5      1/2                 1/2   1/2      1/2
[7 x  + 6 x + 2, -1/5 arctan(------) 5    + 1/5 arctan(2 5   ) 5   , 2 10   ,
                               5

         |           1/2 |
         |         30    |           1/2
      ln(| - 1/2 + ----- |) + ln(2 10   ) + 1
         |          10   |
    - ---------------------------------------, 15.607845206093606153]
          |         1/2 |
          |       30    |           1/2
       ln(| 1/2 + ----- |) + ln(2 10   ) + 1
          |        10   |





    2                         1/2   1/2                 1/2   1/2
[7 x  + 7 x + 2, -2/7 arctan(7   ) 7    + 2/7 arctan(3 7   ) 7   , 7,

         |              1/2 |
         |          2 32    |
      ln(| - 11/7 + ------- |) + ln(7) + 1
         |             7    |
    - ------------------------------------, 26.771696953576777408]
          |            1/2 |
          |        2 32    |
       ln(| 11/7 + ------- |) + ln(7) + 1
          |           7    |





                                                                 1/2
    2                 1/2            1/2         1/2         11 2         1/2
[7 x  + 8 x + 2, 1/2 2    arccoth(2 2   ) - 1/2 2    arccoth(-------), 4 2   ,
                                                                2

          |         1/2 |
          |       34    |          1/2
       ln(| 3/2 - ----- |) + ln(4 2   ) + 1
          |         4   |
    - --------------------------------------, 9.8568612074158945874]
         |           1/2 |
         |         34    |          1/2
      ln(| - 3/2 - ----- |) + ln(4 2   ) + 1
         |           4   |





                                                           1/2
    2                           1/2   1/2              13 2      1/2     1/2
[9 x  + 8 x + 2, -1/2 arctan(2 2   ) 2    + 1/2 arctan(-------) 2   , 4 2   ,
                                                          2

         |           1/2 |
         |         38    |          1/2
      ln(| - 3/2 + ----- |) + ln(4 2   ) + 1
         |           4   |
    - --------------------------------------, 9.3808596317363403541]
          |         1/2 |
          |       38    |          1/2
       ln(| 3/2 + ----- |) + ln(4 2   ) + 1
          |         4   |





                       1/2                 1/2
  2              2 Pi 3                 5 3      1/2   1/2
[x  + 3 x + 3, - --------- + 2/3 arctan(------) 3   , 3   ,
                     9                    3

         |          1/2 |
         |      2 21    |        1/2
      ln(| -3 + ------- |) + ln(3   ) + 1
         |         3    |
    - -----------------------------------, 3.4812632908955237101]
         |         1/2 |
         |     2 21    |        1/2
      ln(| 3 + ------- |) + ln(3   ) + 1
         |        3    |





                                1/2
    2                        2 5      1/2               1/2   1/2      1/2
[3 x  + 4 x + 3, -1/5 arctan(------) 5    + 1/5 arctan(5   ) 5   , 2 10   ,
                               5

         |           1/2 |
         |         30    |           1/2
      ln(| - 1/2 + ----- |) + ln(2 10   ) + 1
         |          10   |
    - ---------------------------------------, 15.607845206093606153]
          |         1/2 |
          |       30    |           1/2
       ln(| 1/2 + ----- |) + ln(2 10   ) + 1
          |        10   |





                                  1/2
    2                         5 11       1/2                 1/2    1/2    1/2
[3 x  + 5 x + 3, -2/11 arctan(-------) 11    + 2/11 arctan(11   ) 11   , 11   ,
                                11

         |          1/2 |
         |      2 33    |         1/2
      ln(| -1 + ------- |) + ln(11   ) + 1
         |        11    |
    - ------------------------------------, 4.1879821595890442019]
         |         1/2 |
         |     2 33    |         1/2
      ln(| 1 + ------- |) + ln(11   ) + 1
         |       11    |





                                                              1/2
    2                         1/2   1/2        1/2         5 3         1/2
[2 x  + 6 x + 3, 1/3 arccoth(3   ) 3    - 1/3 3    arccoth(------), 4 3   ,
                                                             3

         |       1/2 |
         |     33    |          1/2
      ln(| 1 - ----- |) + ln(4 3   ) + 1
         |       6   |
    - -----------------------------------, 17.326964322421861472]
         |        1/2 |
         |      33    |          1/2
      ln(| -1 - ----- |) + ln(4 3   ) + 1
         |        6   |





                       1/2                 1/2
    2              Pi 3                 7 3      1/2     1/2
[4 x  + 6 x + 3, - ------- + 1/3 arctan(------) 3   , 4 3   ,
                      9                   3

         |        1/2 |
         |      39    |          1/2
      ln(| -1 + ----- |) + ln(4 3   ) + 1
         |        6   |
    - -----------------------------------, 14.892104575824395585]
         |       1/2 |
         |     39    |          1/2
      ln(| 1 + ----- |) + ln(4 3   ) + 1
         |       6   |





                              1/2                       1/2
    2                        3      1/2              5 3      1/2     1/2
[7 x  + 6 x + 3, -1/6 arctan(----) 3    + 1/6 arctan(------) 3   , 4 3   ,
                              2                        3

         |           1/2 |
         |         48    |          1/2
      ln(| - 1/4 + ----- |) + ln(4 3   ) + 1
         |          24   |
    - --------------------------------------, 8.3099863401554735279]
          |         1/2 |
          |       48    |          1/2
       ln(| 1/4 + ----- |) + ln(4 3   ) + 1
          |        24   |





                                  1/2
    2                         7 13       1/2          1/2           1/2     1/2
[3 x  + 7 x + 3, 2/13 arccoth(-------) 13    - 2/13 13    arccoth(13   ), 13   ,
                                13

         |         1/2 |
         |     2 39    |         1/2
      ln(| 1 - ------- |) + ln(13   ) + 1
         |       13    |
    - ------------------------------------, 4.0924630490843634450]
         |          1/2 |
         |      2 39    |         1/2
      ln(| -1 - ------- |) + ln(13   ) + 1
         |        13    |





    2                           1/2   1/2                 1/2   1/2
[6 x  + 8 x + 3, -1/2 arctan(2 2   ) 2    + 1/2 arctan(5 2   ) 2   , 8,

         |           1/2 |
         |         51    |
      ln(| - 7/4 + ----- |) + ln(8) + 1
         |           4   |
    - ---------------------------------, 17.522143636180518454]
          |         1/2 |
          |       51    |
       ln(| 7/4 + ----- |) + ln(8) + 1
          |         4   |





                                                           1/2
    2                           1/2   1/2              23 3      1/2   1/2
[7 x  + 9 x + 3, -2/3 arctan(3 3   ) 3    + 2/3 arctan(-------) 3   , 3   ,
                                                          3

         |          1/2 |
         |      2 57    |        1/2
      ln(| -5 + ------- |) + ln(3   ) + 1
         |         3    |
    - -----------------------------------, 3.0780462199168187258]
         |         1/2 |
         |     2 57    |        1/2
      ln(| 5 + ------- |) + ln(3   ) + 1
         |        3    |





                                  1/2                          1/2
    2                         3 15       1/2               5 15       1/2
[8 x  + 9 x + 3, -2/15 arctan(-------) 15    + 2/15 arctan(-------) 15   ,
                                 5                            3

                |          1/2 |
                |      2 60    |         1/2
             ln(| -1 + ------- |) + ln(15   ) + 1
      1/2       |        15    |
    15   , - ------------------------------------, 3.8806942681404861577]
                |         1/2 |
                |     2 60    |         1/2
             ln(| 1 + ------- |) + ln(15   ) + 1
                |       15    |





                     1/2                 1/2
  2              Pi 3                 2 3      1/2
[x  + 2 x + 4, - ------- + 1/3 arctan(------) 3   , 6,
                   18                   3

         |           1/2 |
         |         28    |
      ln(| - 5/6 + ----- |) + ln(6) + 1
         |           6   |
    - ---------------------------------, 15.315550900340627783]
          |         1/2 |
          |       28    |
       ln(| 5/6 + ----- |) + ln(6) + 1
          |         6   |





                              1/2                       1/2
  2                        3 7      1/2              5 7      1/2
[x  + 3 x + 4, -2/7 arctan(------) 7    + 2/7 arctan(------) 7   , 7,
                             7                         7

         |              1/2 |
         |          2 32    |
      ln(| - 11/7 + ------- |) + ln(7) + 1
         |             7    |
    - ------------------------------------, 26.771696953576777408]
          |            1/2 |
          |        2 32    |
       ln(| 11/7 + ------- |) + ln(7) + 1
          |           7    |





                                1/2                       1/2
    2                        5 7      1/2              9 7      1/2
[2 x  + 5 x + 4, -2/7 arctan(------) 7    + 2/7 arctan(------) 7   , 7,
                               7                         7

         |              1/2 |
         |          2 44    |
      ln(| - 13/7 + ------- |) + ln(7) + 1
         |             7    |
    - ------------------------------------, 14.239775923153759269]
          |            1/2 |
          |        2 44    |
       ln(| 13/7 + ------- |) + ln(7) + 1
          |           7    |





                                   1/2                        1/2
  2                 1/2         3 5            1/2         4 5
[x  + 6 x + 4, 1/5 5    arccoth(------) - 1/5 5    arccoth(------), 10,
                                  5                          5

          |          1/2 |
          |        44    |
       ln(| 7/10 - ----- |) + ln(10) + 1
          |         10   |
    - -----------------------------------, 1176.2902524845005445]
         |            1/2 |
         |          44    |
      ln(| - 7/10 - ----- |) + ln(10) + 1
         |           10   |





                       1/2
    2              Pi 3                    1/2   1/2
[3 x  + 6 x + 4, - ------- + 1/3 arctan(2 3   ) 3   , 6,
                      9

         |           1/2 |
         |         52    |
      ln(| - 7/6 + ----- |) + ln(6) + 1
         |           6   |
    - ---------------------------------, 7.5789513504094747847]
          |         1/2 |
          |       52    |
       ln(| 7/6 + ----- |) + ln(6) + 1
          |         6   |





                                  1/2
    2                         7 15       1/2                 1/2    1/2    1/2
[4 x  + 7 x + 4, -2/15 arctan(-------) 15    + 2/15 arctan(15   ) 15   , 15   ,
                                15

         |          1/2 |
         |      2 60    |         1/2
      ln(| -1 + ------- |) + ln(15   ) + 1
         |        15    |
    - ------------------------------------, 3.8806942681404861577]
         |         1/2 |
         |     2 60    |         1/2
      ln(| 1 + ------- |) + ln(15   ) + 1
         |       15    |





    2
[5 x  + 8 x + 4, -1/2 arctan(2) + 1/2 arctan(9/2), 8,

         |        1/2 |
         |      68    |
      ln(| -1 + ----- |) + ln(8) + 1
         |        8   |
    - ------------------------------, 10.432735096002347775]
         |       1/2 |
         |     68    |
      ln(| 1 + ----- |) + ln(8) + 1
         |       8   |





                                  1/2
    2                         9 17       1/2          1/2           1/2     1/2
[4 x  + 9 x + 4, 2/17 arccoth(-------) 17    - 2/17 17    arccoth(17   ), 17   ,
                                17

         |         1/2 |
         |     2 68    |         1/2
      ln(| 1 - ------- |) + ln(17   ) + 1
         |       17    |
    - ------------------------------------, 3.8269636216410127965]
         |          1/2 |
         |      2 68    |         1/2
      ln(| -1 - ------- |) + ln(17   ) + 1
         |        17    |





    2                  1/2          1/2         1/2            1/2
[5 x  + 10 x + 4, 1/5 5    arccoth(5   ) - 1/5 5    arccoth(2 5   ), 10,

          |          1/2 |
          |        76    |
       ln(| 9/10 - ----- |) + ln(10) + 1
          |         10   |
    - -----------------------------------, 15.613751960540034505]
         |            1/2 |
         |          76    |
      ln(| - 9/10 - ----- |) + ln(10) + 1
         |           10   |





                                 1/2
    2                         5 3      1/2                 1/2   1/2     1/2
[7 x  + 10 x + 4, -1/3 arctan(------) 3    + 1/3 arctan(4 3   ) 3   , 2 3   ,
                                3

         |           1/2 |
         |         84    |          1/2
      ln(| - 3/2 + ----- |) + ln(2 3   ) + 1
         |           6   |
    - --------------------------------------, 3.4812632908955237133]
          |         1/2 |
          |       84    |          1/2
       ln(| 3/2 + ----- |) + ln(2 3   ) + 1
          |         6   |





  2                                Pi      1/2
[x  + 2 x + 5, -1/2 arctan(1/2) + ----, 4 2   ,
                                   8

         |           1/2 |
         |         40    |          1/2
      ln(| - 3/4 + ----- |) + ln(4 2   ) + 1
         |           8   |
    - --------------------------------------, 7.7073575485293709834]
          |         1/2 |
          |       40    |          1/2
       ln(| 3/4 + ----- |) + ln(4 2   ) + 1
          |         8   |





                                                            1/2
  2                 1/2          1/2         1/2         7 5       1/2
[x  + 5 x + 5, 2/5 5    arccoth(5   ) - 2/5 5    arccoth(------), 5   ,
                                                           5

         |         1/2 |
         |     2 55    |        1/2
      ln(| 3 - ------- |) + ln(5   ) + 1
         |        5    |
    - -----------------------------------, 3.2571690113448376880]
         |          1/2 |
         |      2 55    |        1/2
      ln(| -3 - ------- |) + ln(5   ) + 1
         |         5    |





                                1/2                          1/2
    2                         15       1/2               3 15       1/2    1/2
[2 x  + 5 x + 5, -2/15 arctan(-----) 15    + 2/15 arctan(-------) 15   , 15   ,
                                3                           5

         |          1/2 |
         |      2 60    |         1/2
      ln(| -1 + ------- |) + ln(15   ) + 1
         |        15    |
    - ------------------------------------, 3.8806942681404861577]
         |         1/2 |
         |     2 60    |         1/2
      ln(| 1 + ------- |) + ln(15   ) + 1
         |       15    |





                                                 |        1/2 |
                                                 |      65    |
                                              ln(| -4 + ----- |) + ln(4) + 1
    2                                            |        2   |
[2 x  + 6 x + 5, -arctan(3) + arctan(5), 4, - ------------------------------,
                                                 |       1/2 |
                                                 |     65    |
                                              ln(| 4 + ----- |) + ln(4) + 1
                                                 |       2   |

    5.1258339353606460401]





                                  1/2                           1/2
    2                         7 11       1/2               13 11       1/2
[3 x  + 7 x + 5, -2/11 arctan(-------) 11    + 2/11 arctan(--------) 11   , 11,
                                11                            11

         |            1/2 |
         |   17   2 75    |
      ln(| - -- + ------- |) + ln(11) + 1
         |   11     11    |
    - -----------------------------------, 33.903991854273836881]
          |          1/2 |
          | 17   2 75    |
       ln(| -- + ------- |) + ln(11) + 1
          | 11     11    |





                                  1/2                        1/2
    2                         7 51       1/2               51       1/2
[5 x  + 7 x + 5, -2/51 arctan(-------) 51    + 2/51 arctan(-----) 51   ,
                                51                           3

                  |             1/2 |
                  |         2 85    |           1/2
               ln(| - 1/3 + ------- |) + ln(3 17   ) + 1
        1/2       |           51    |
    3 17   , - -----------------------------------------, 60.912947227427311286
                   |           1/2 |
                   |       2 85    |           1/2
                ln(| 1/3 + ------- |) + ln(3 17   ) + 1
                   |         51    |

    ]





                                  1/2
    2                         9 19       1/2                 1/2    1/2    1/2
[5 x  + 9 x + 5, -2/19 arctan(-------) 19    + 2/19 arctan(19   ) 19   , 19   ,
                                19

         |          1/2 |
         |      2 95    |         1/2
      ln(| -1 + ------- |) + ln(19   ) + 1
         |        19    |
    - ------------------------------------, 3.6974006367249319472]
         |         1/2 |
         |     2 95    |         1/2
      ln(| 1 + ------- |) + ln(19   ) + 1
         |       19    |





                              1/2                              1/2
  2                          5      1/2         1/2         3 5         1/2
[x  + 10 x + 5, 1/10 arccoth(----) 5    - 1/10 5    arccoth(------), 4 5   ,
                              2                               5

          |         1/2 |
          |       80    |          1/2
       ln(| 1/4 - ----- |) + ln(4 5   ) + 1
          |        40   |
    - --------------------------------------, 6.5082057497631049136]
         |           1/2 |
         |         80    |          1/2
      ln(| - 1/4 - ----- |) + ln(4 5   ) + 1
         |          40   |





                                       1/2                           1/2
    2                    1/2         10              1/2         4 10
[3 x  + 10 x + 5, 1/10 10    arccoth(-----) - 1/10 10    arccoth(-------),
                                       2                            5

                   |         1/2 |
                   |       90    |           1/2
                ln(| 1/2 - ----- |) + ln(4 10   ) + 1
        1/2        |        20   |
    4 10   , - ---------------------------------------, 29.025649285508016267]
                  |           1/2 |
                  |         90    |           1/2
               ln(| - 1/2 - ----- |) + ln(4 10   ) + 1
                  |          20   |





                                                          1/2
    2                  1/2          1/2                9 5      1/2     1/2
[4 x  + 10 x + 5, 1/5 5    arccoth(5   ) - 1/5 arccoth(------) 5   , 4 5   ,
                                                         5

         |       1/2 |
         |     95    |          1/2
      ln(| 1 - ----- |) + ln(4 5   ) + 1
         |      10   |
    - -----------------------------------, 8.9802507673707511949]
         |        1/2 |
         |      95    |          1/2
      ln(| -1 - ----- |) + ln(4 5   ) + 1
         |       10   |





                                                          1/2
    2                          1/2   1/2              11 5      1/2     1/2
[6 x  + 10 x + 5, -1/5 arctan(5   ) 5    + 1/5 arctan(-------) 5   , 4 5   ,
                                                         5

         |         1/2 |
         |      105    |          1/2
      ln(| -1 + ------ |) + ln(4 5   ) + 1
         |        10   |
    - ------------------------------------, 8.6379958426045502068]
         |        1/2 |
         |     105    |          1/2
      ln(| 1 + ------ |) + ln(4 5   ) + 1
         |       10   |





                                 1/2                          1/2
    2                          10       1/2               6 10       1/2
[7 x  + 10 x + 5, -1/10 arctan(-----) 10    + 1/10 arctan(-------) 10   ,
                                 2                           5

                  |            1/2 |
                  |         110    |           1/2
               ln(| - 1/2 + ------ |) + ln(4 10   ) + 1
        1/2       |           20   |
    4 10   , - ----------------------------------------, 21.305691483920764832]
                   |          1/2 |
                   |       110    |           1/2
                ln(| 1/2 + ------ |) + ln(4 10   ) + 1
                   |         20   |





                                1/2                        1/2
    2                          5      1/2               7 5      1/2      1/2
[9 x  + 10 x + 5, -1/10 arctan(----) 5    + 1/10 arctan(------) 5   , 4 10   ,
                                2                         5

         |            1/2 |
         |         120    |           1/2
      ln(| - 1/4 + ------ |) + ln(4 10   ) + 1
         |           40   |
    - ----------------------------------------, 15.607845206093606123]
          |          1/2 |
          |       120    |           1/2
       ln(| 1/4 + ------ |) + ln(4 10   ) + 1
          |         40   |





                              1/2                        1/2
  2                         15       1/2               15       1/2    1/2
[x  + 3 x + 6, -2/15 arctan(-----) 15    + 2/15 arctan(-----) 15   , 15   ,
                              5                          3

         |          1/2 |
         |      2 60    |         1/2
      ln(| -1 + ------- |) + ln(15   ) + 1
         |        15    |
    - ------------------------------------, 3.8806942681404861577]
         |         1/2 |
         |     2 60    |         1/2
      ln(| 1 + ------- |) + ln(15   ) + 1
         |       15    |





                                                       1/2
  2                         1/2   1/2               4 3      1/2     1/2
[x  + 6 x + 6, 1/3 arccoth(3   ) 3    - 1/3 arccoth(------) 3   , 2 6   ,
                                                      3

          |         1/2 |
          |       78    |          1/2
       ln(| 3/2 - ----- |) + ln(2 6   ) + 1
          |         6   |
    - --------------------------------------, 4.7338779958071950700]
         |           1/2 |
         |         78    |          1/2
      ln(| - 3/2 - ----- |) + ln(2 6   ) + 1
         |           6   |





                                      1/2                         1/2
  2                   1/2         2 10              1/2         10          1/2
[x  + 8 x + 6, 1/10 10    arccoth(-------) - 1/10 10    arccoth(-----), 4 10   ,
                                     5                            2

          |         1/2 |
          |       90    |           1/2
       ln(| 1/2 - ----- |) + ln(4 10   ) + 1
          |        20   |
    - ---------------------------------------, 29.025649285508016267]
         |           1/2 |
         |         90    |           1/2
      ln(| - 1/2 - ----- |) + ln(4 10   ) + 1
         |          20   |





                                                          1/2
    2                           1/2   1/2              7 2      1/2     1/2
[3 x  + 8 x + 6, -1/2 arctan(2 2   ) 2    + 1/2 arctan(------) 2   , 4 2   ,
                                                         2

         |            1/2 |
         |         102    |          1/2
      ln(| - 5/2 + ------ |) + ln(4 2   ) + 1
         |           4    |
    - ---------------------------------------, 5.5238132377130025906]
          |          1/2 |
          |       102    |          1/2
       ln(| 5/2 + ------ |) + ln(4 2   ) + 1
          |         4    |





                                  1/2                           1/2
    2                         3 15       1/2               17 15       1/2
[4 x  + 9 x + 6, -2/15 arctan(-------) 15    + 2/15 arctan(--------) 15   ,
                                 5                            15

                 |              1/2 |
                 |         2 114    |          1/2
              ln(| - 7/5 + -------- |) + ln(5 3   ) + 1
       1/2       |            15    |
    5 3   , - -----------------------------------------, 8.1455807933567691820]
                  |            1/2 |
                  |       2 114    |          1/2
               ln(| 7/5 + -------- |) + ln(5 3   ) + 1
                  |          15    |





    2                          1/2   1/2                 1/2   1/2      1/2
[5 x  + 10 x + 6, -1/5 arctan(5   ) 5    + 1/5 arctan(2 5   ) 5   , 10 2   ,

         |           1/2 |
         |   11   126    |           1/2
      ln(| - -- + ------ |) + ln(10 2   ) + 1
         |   10     10   |
    - ---------------------------------------, 31.631103067531731805]
          |         1/2 |
          | 11   126    |           1/2
       ln(| -- + ------ |) + ln(10 2   ) + 1
          | 10     10   |





                          1/2             1/2
  2                      3      1/2   Pi 3
[x  + x + 7, -2/9 arctan(----) 3    + -------, 9,
                          9             27

         |             1/2 |
         |         2 63    |
      ln(| - 5/9 + ------- |) + ln(9) + 1
         |           27    |
    - -----------------------------------, 15.315550900340627740]
          |           1/2 |
          |       2 63    |
       ln(| 5/9 + ------- |) + ln(9) + 1
          |         27    |





                              1/2             1/2
  2                        2 3      1/2   Pi 3        1/2
[x  + 4 x + 7, -1/3 arctan(------) 3    + -------, 2 3   ,
                             3               9

         |           1/2 |
         |         84    |          1/2
      ln(| - 3/2 + ----- |) + ln(2 3   ) + 1
         |           6   |
    - --------------------------------------, 3.4812632908955237133]
          |         1/2 |
          |       84    |          1/2
       ln(| 3/2 + ----- |) + ln(2 3   ) + 1
          |         6   |





                              1/2                       1/2
  2                        5 3      1/2              7 3      1/2
[x  + 5 x + 7, -2/3 arctan(------) 3    + 2/3 arctan(------) 3   , 3,
                             3                         3

         |              1/2 |
         |          2 91    |
      ln(| - 19/3 + ------- |) + ln(3) + 1
         |             3    |
    - ------------------------------------, 4.0107214983056850938]
          |            1/2 |
          |        2 91    |
       ln(| 19/3 + ------- |) + ln(3) + 1
          |           3    |





                                   1/2
  2                 1/2         3 2            1/2            1/2      1/2
[x  + 6 x + 7, 1/2 2    arccoth(------) - 1/2 2    arccoth(2 2   ), 4 2   ,
                                  2

          |         1/2 |
          |       98    |          1/2
       ln(| 5/2 - ----- |) + ln(4 2   ) + 1
          |         4   |
    - --------------------------------------, 5.5608711225648792435]
         |           1/2 |
         |         98    |          1/2
      ln(| - 5/2 - ----- |) + ln(4 2   ) + 1
         |           4   |





                                1/2
    2                        3 5      1/2               1/2   1/2     1/2
[2 x  + 6 x + 7, -1/5 arctan(------) 5    + 1/5 arctan(5   ) 5   , 4 5   ,
                               5

         |         1/2 |
         |      105    |          1/2
      ln(| -1 + ------ |) + ln(4 5   ) + 1
         |        10   |
    - ------------------------------------, 8.6379958426045502068]
         |        1/2 |
         |     105    |          1/2
      ln(| 1 + ------ |) + ln(4 5   ) + 1
         |       10   |





                              1/2             1/2
    2                        3      1/2   Pi 3
[3 x  + 6 x + 7, -1/6 arctan(----) 3    + -------, 12,
                              2             18

         |             1/2 |
         |          112    |
      ln(| - 5/12 + ------ |) + ln(12) + 1
         |            24   |
    - ------------------------------------, 15.315550900340627746]
          |           1/2 |
          |        112    |
       ln(| 5/12 + ------ |) + ln(12) + 1
          |          24   |





                                    1/2                           1/2
  2                   1/2         21              1/2         3 21        1/2
[x  + 7 x + 7, 2/21 21    arccoth(-----) - 2/21 21    arccoth(-------), 21   ,
                                    3                            7

         |          1/2 |
         |     2 105    |         1/2
      ln(| 1 - -------- |) + ln(21   ) + 1
         |        21    |
    - -------------------------------------, 3.6621162818653647041]
         |           1/2 |
         |      2 105    |         1/2
      ln(| -1 - -------- |) + ln(21   ) + 1
         |         21    |





                                                         1/2
    2                         1/2   1/2              11 7      1/2   1/2
[2 x  + 7 x + 7, -2/7 arctan(7   ) 7    + 2/7 arctan(-------) 7   , 7   ,
                                                        7

         |           1/2 |
         |      2 112    |        1/2
      ln(| -3 + -------- |) + ln(7   ) + 1
         |         7     |
    - ------------------------------------, 3.1307999826699309253]
         |          1/2 |
         |     2 112    |        1/2
      ln(| 3 + -------- |) + ln(7   ) + 1
         |        7     |





                                1/2                           1/2
    2                         35       1/2               13 35       1/2
[3 x  + 7 x + 7, -2/35 arctan(-----) 35    + 2/35 arctan(--------) 35   ,
                                5                           35

                 |              1/2 |
                 |         2 119    |          1/2
              ln(| - 3/5 + -------- |) + ln(5 7   ) + 1
       1/2       |            35    |
    5 7   , - -----------------------------------------, 22.678279240725504417]
                  |            1/2 |
                  |       2 119    |          1/2
               ln(| 3/5 + -------- |) + ln(5 7   ) + 1
                  |          35    |





                               1/2                        1/2
    2                         7      1/2               5 7      1/2     1/2
[4 x  + 7 x + 7, -2/21 arctan(----) 7    + 2/21 arctan(------) 7   , 3 7   ,
                               3                         7

         |              1/2 |
         |         2 126    |          1/2
      ln(| - 1/3 + -------- |) + ln(3 7   ) + 1
         |            63    |
    - -----------------------------------------, 4.8569705938289177923]
          |            1/2 |
          |       2 126    |          1/2
       ln(| 1/3 + -------- |) + ln(3 7   ) + 1
          |          63    |





    2                 1/2            1/2                   1/2   1/2
[2 x  + 8 x + 7, 1/2 2    arccoth(2 2   ) - 1/2 arccoth(3 2   ) 2   , 8,

          |           1/2 |
          |        119    |
       ln(| 11/4 - ------ |) + ln(8) + 1
          |          4    |
    - -----------------------------------, 7.8228657034259635161]
         |             1/2 |
         |          119    |
      ln(| - 11/4 - ------ |) + ln(8) + 1
         |            4    |





                                1/2                       1/2
    2                        4 5      1/2              7 5      1/2      1/2
[3 x  + 8 x + 7, -1/5 arctan(------) 5    + 1/5 arctan(------) 5   , 10 2   ,
                               5                         5

         |           1/2 |
         |   11   126    |           1/2
      ln(| - -- + ------ |) + ln(10 2   ) + 1
         |   10     10   |
    - ---------------------------------------, 31.631103067531731805]
          |         1/2 |
          | 11   126    |           1/2
       ln(| -- + ------ |) + ln(10 2   ) + 1
          | 10     10   |





    2                           1/2   1/2                 1/2   1/2
[3 x  + 9 x + 7, -2/3 arctan(3 3   ) 3    + 2/3 arctan(5 3   ) 3   , 3,

         |               1/2 |
         |          2 133    |
      ln(| - 23/3 + -------- |) + ln(3) + 1
         |             3     |
    - -------------------------------------, 3.7896117451863288465]
          |             1/2 |
          |        2 133    |
       ln(| 23/3 + -------- |) + ln(3) + 1
          |           3     |





                                 1/2
    2                         5 3      1/2                 1/2   1/2     1/2
[4 x  + 10 x + 7, -1/3 arctan(------) 3    + 1/3 arctan(3 3   ) 3   , 4 3   ,
                                3

         |         1/2 |
         |      147    |          1/2
      ln(| -2 + ------ |) + ln(4 3   ) + 1
         |        6    |
    - ------------------------------------, 5.5994975951404949745]
         |        1/2 |
         |     147    |          1/2
      ln(| 2 + ------ |) + ln(4 3   ) + 1
         |       6    |





  2               Pi                        1/2
[x  + 4 x + 8, - ---- + 1/2 arctan(3/2), 4 2   ,
                  8

         |            1/2 |
         |         104    |          1/2
      ln(| - 5/4 + ------ |) + ln(4 2   ) + 1
         |           8    |
    - ---------------------------------------, 4.7883274317566198544]
          |          1/2 |
          |       104    |          1/2
       ln(| 5/4 + ------ |) + ln(4 2   ) + 1
          |         8    |





                               1/2                      1/2
    2                         5      1/2               5      1/2      1/2
[3 x  + 4 x + 8, -1/10 arctan(----) 5    + 1/10 arctan(----) 5   , 4 10   ,
                               5                        2

         |            1/2 |
         |         120    |           1/2
      ln(| - 1/4 + ------ |) + ln(4 10   ) + 1
         |           40   |
    - ----------------------------------------, 15.607845206093606123]
          |          1/2 |
          |       120    |           1/2
       ln(| 1/4 + ------ |) + ln(4 10   ) + 1
          |         40   |





                              1/2
  2                        5 7      1/2               1/2   1/2   1/2
[x  + 5 x + 8, -2/7 arctan(------) 7    + 2/7 arctan(7   ) 7   , 7   ,
                             7

         |           1/2 |
         |      2 112    |        1/2
      ln(| -3 + -------- |) + ln(7   ) + 1
         |         7     |
    - ------------------------------------, 3.1307999826699309253]
         |          1/2 |
         |     2 112    |        1/2
      ln(| 3 + -------- |) + ln(7   ) + 1
         |        7     |





                                  1/2                           1/2
    2                         7 15       1/2               11 15       1/2
[2 x  + 7 x + 8, -2/15 arctan(-------) 15    + 2/15 arctan(--------) 15   , 15,
                                15                            15

         |             1/2 |
         |   23   2 136    |
      ln(| - -- + -------- |) + ln(15) + 1
         |   15      15    |
    - ------------------------------------, 38.895259299026773802]
          |           1/2 |
          | 23   2 136    |
       ln(| -- + -------- |) + ln(15) + 1
          | 15      15    |





                                                       1/2
  2                 1/2          1/2                5 2      1/2     1/2
[x  + 8 x + 8, 1/4 2    arccoth(2   ) - 1/4 arccoth(------) 2   , 8 2   ,
                                                      4

          |          1/2 |
          |       136    |          1/2
       ln(| 3/4 - ------ |) + ln(8 2   ) + 1
          |         16   |
    - ---------------------------------------, 9.8568612074158945565]
         |            1/2 |
         |         136    |          1/2
      ln(| - 3/4 - ------ |) + ln(8 2   ) + 1
         |           16   |





                                                        1/2
    2                         1/2   1/2              7 2      1/2     1/2
[3 x  + 8 x + 8, -1/4 arctan(2   ) 2    + 1/4 arctan(------) 2   , 8 2   ,
                                                       4

         |            1/2 |
         |         152    |          1/2
      ln(| - 3/4 + ------ |) + ln(8 2   ) + 1
         |           16   |
    - ---------------------------------------, 9.3808596317363403614]
          |          1/2 |
          |       152    |          1/2
       ln(| 3/4 + ------ |) + ln(8 2   ) + 1
          |         16   |





                                  1/2                                  1/2
    2                         9 17       1/2          1/2         13 17
[2 x  + 9 x + 8, 2/17 arccoth(-------) 17    - 2/17 17    arccoth(--------), 17,
                                17                                   17

          |           1/2 |
          | 25   2 152    |
       ln(| -- - -------- |) + ln(17) + 1
          | 17      17    |
    - ------------------------------------, 69.184664932665641876]
         |             1/2 |
         |   25   2 152    |
      ln(| - -- - -------- |) + ln(17) + 1
         |   17      17    |





                                  1/2
    2                         3 15       1/2                 1/2    1/2     1/2
[3 x  + 9 x + 8, -2/15 arctan(-------) 15    + 2/15 arctan(15   ) 15   , 3 5   ,
                                 5

         |              1/2 |
         |         2 160    |          1/2
      ln(| - 5/3 + -------- |) + ln(3 5   ) + 1
         |            15    |
    - -----------------------------------------, 5.0539077534823343831]
          |            1/2 |
          |       2 160    |          1/2
       ln(| 5/3 + -------- |) + ln(3 5   ) + 1
          |          15    |





                     1/2                 1/2
  2              Pi 3                 5 3      1/2
[x  + 3 x + 9, - ------- + 2/9 arctan(------) 3   , 9,
                   27                   9

         |              1/2 |
         |         2 117    |
      ln(| - 7/9 + -------- |) + ln(9) + 1
         |            27    |
    - ------------------------------------, 7.5789513504094747810]
          |            1/2 |
          |       2 117    |
       ln(| 7/9 + -------- |) + ln(9) + 1
          |          27    |





                               1/2                      1/2
    2                         7      1/2               7      1/2     1/2
[2 x  + 3 x + 9, -2/21 arctan(----) 7    + 2/21 arctan(----) 7   , 3 7   ,
                               7                        3

         |              1/2 |
         |         2 126    |          1/2
      ln(| - 1/3 + -------- |) + ln(3 7   ) + 1
         |            63    |
    - -----------------------------------------, 4.8569705938289177923]
          |            1/2 |
          |       2 126    |          1/2
       ln(| 1/3 + -------- |) + ln(3 7   ) + 1
          |          63    |





                              1/2                       1/2
  2                        2 5      1/2              3 5      1/2      1/2
[x  + 4 x + 9, -1/5 arctan(------) 5    + 1/5 arctan(------) 5   , 10 2   ,
                             5                         5

         |           1/2 |
         |   11   126    |           1/2
      ln(| - -- + ------ |) + ln(10 2   ) + 1
         |   10     10   |
    - ---------------------------------------, 31.631103067531731805]
          |         1/2 |
          | 11   126    |           1/2
       ln(| -- + ------ |) + ln(10 2   ) + 1
          | 10     10   |





                                1/2                          1/2
  2                         5 11       1/2               7 11       1/2
[x  + 5 x + 9, -2/11 arctan(-------) 11    + 2/11 arctan(-------) 11   , 11,
                              11                           11

         |             1/2 |
         |   23   2 135    |
      ln(| - -- + -------- |) + ln(11) + 1
         |   11      11    |
    - ------------------------------------, 12.089124796145223893]
          |           1/2 |
          | 23   2 135    |
       ln(| -- + -------- |) + ln(11) + 1
          | 11      11    |





    2               Pi
[2 x  + 6 x + 9, - ---- + 1/3 arctan(5/3), 12,
                    12

         |            1/2 |
         |         153    |
      ln(| - 2/3 + ------ |) + ln(12) + 1
         |           18   |
    - -----------------------------------, 10.432735096002347784]
          |          1/2 |
          |       153    |
       ln(| 2/3 + ------ |) + ln(12) + 1
          |         18   |





                                1/2                           1/2
  2                         7 13       1/2                9 13       1/2
[x  + 7 x + 9, 2/13 arccoth(-------) 13    - 2/13 arccoth(-------) 13   , 13,
                              13                            13

          |           1/2 |
          | 25   2 153    |
       ln(| -- - -------- |) + ln(13) + 1
          | 13      13    |
    - ------------------------------------, 15.354341238420183413]
         |             1/2 |
         |   25   2 153    |
      ln(| - -- - -------- |) + ln(13) + 1
         |   13      13    |





    2                           1/2   1/2                 1/2   1/2
[2 x  + 8 x + 9, -1/2 arctan(2 2   ) 2    + 1/2 arctan(3 2   ) 2   , 8,

         |             1/2 |
         |          171    |
      ln(| - 13/4 + ------ |) + ln(8) + 1
         |            4    |
    - -----------------------------------, 6.6635635425939131729]
          |           1/2 |
          |        171    |
       ln(| 13/4 + ------ |) + ln(8) + 1
          |          4    |





                                    1/2                     1/2
  2                  1/2         3 5                    11 5      1/2
[x  + 9 x + 9, 2/15 5    arccoth(------) - 2/15 arccoth(-------) 5   , 15,
                                   5                      15

          |            1/2 |
          |       2 171    |
       ln(| 3/5 - -------- |) + ln(15) + 1
          |          45    |
    - -------------------------------------, 15.613751960540034478]
         |              1/2 |
         |         2 171    |
      ln(| - 3/5 - -------- |) + ln(15) + 1
         |            45    |





                                 1/2
    2                         5 7      1/2        1/2          1/2      1/2
[2 x  + 10 x + 9, 1/7 arccoth(------) 7    - 1/7 7    arccoth(7   ), 4 7   ,
                                7

         |        1/2 |
         |     189    |          1/2
      ln(| 1 - ------ |) + ln(4 7   ) + 1
         |       14   |
    - ------------------------------------, 7.1537717511215774039]
         |         1/2 |
         |      189    |          1/2
      ln(| -1 - ------ |) + ln(4 7   ) + 1
         |        14   |





                                 1/2
    2                         5 2      1/2                 1/2   1/2     1/2
[3 x  + 10 x + 9, -1/2 arctan(------) 2    + 1/2 arctan(4 2   ) 2   , 4 2   ,
                                2

         |            1/2 |
         |         198    |          1/2
      ln(| - 7/2 + ------ |) + ln(4 2   ) + 1
         |           4    |
    - ---------------------------------------, 4.6148485335234683752]
          |          1/2 |
          |       198    |          1/2
       ln(| 7/2 + ------ |) + ln(4 2   ) + 1
          |         4    |





                               1/2                          1/2
  2                          15       1/2               7 15       1/2     1/2
[x  + 5 x + 10, -2/15 arctan(-----) 15    + 2/15 arctan(-------) 15   , 3 5   ,
                               3                          15

         |              1/2 |
         |         2 160    |          1/2
      ln(| - 5/3 + -------- |) + ln(3 5   ) + 1
         |            15    |
    - -----------------------------------------, 5.0539077534823343831]
          |            1/2 |
          |       2 160    |          1/2
       ln(| 5/3 + -------- |) + ln(3 5   ) + 1
          |          15    |





                               1/2                        1/2
  2                         2 6      1/2               5 6      1/2      1/2
[x  + 8 x + 10, 1/6 arccoth(------) 6    - 1/6 arccoth(------) 6   , 12 2   ,
                              3                          6

          |          1/2 |
          |       190    |           1/2
       ln(| 7/6 - ------ |) + ln(12 2   ) + 1
          |         12   |
    - ----------------------------------------, 26.095688442030019397]
         |            1/2 |
         |         190    |           1/2
      ln(| - 7/6 - ------ |) + ln(12 2   ) + 1
         |           12   |





                                   1/2                        1/2
    2                          2 14       1/2               14       1/2
[3 x  + 8 x + 10, -1/14 arctan(-------) 14    + 1/14 arctan(-----) 14   ,
                                  7                           2

                  |            1/2 |
                  |         210    |           1/2
               ln(| - 1/2 + ------ |) + ln(4 14   ) + 1
        1/2       |           28   |
    4 14   , - ----------------------------------------, 12.050569329465964557]
                   |          1/2 |
                   |       210    |           1/2
                ln(| 1/2 + ------ |) + ln(4 14   ) + 1
                   |         28   |





                                                          1/2
    2                           1/2   1/2              8 5      1/2      1/2
[3 x  + 10 x + 10, -1/5 arctan(5   ) 5    + 1/5 arctan(------) 5   , 2 10   ,
                                                         5

         |            1/2 |
         |         230    |           1/2
      ln(| - 3/2 + ------ |) + ln(2 10   ) + 1
         |           10   |
    - ----------------------------------------, 4.1452252161237347194]
          |          1/2 |
          |       230    |           1/2
       ln(| 3/2 + ------ |) + ln(2 10   ) + 1
          |         10   |





                                 1/2                        1/2
    2                           5      1/2               4 5      1/2      1/2
[7 x  + 10 x + 10, -1/15 arctan(----) 5    + 1/15 arctan(------) 5   , 6 10   ,
                                 3                         5

         |            1/2 |
         |         270    |           1/2
      ln(| - 1/6 + ------ |) + ln(6 10   ) + 1
         |           90   |
    - ----------------------------------------, 15.607845206093606123]
          |          1/2 |
          |       270    |           1/2
       ln(| 1/6 + ------ |) + ln(6 10   ) + 1
          |         90   |



                              and in Maple format



[x^2+x+1, 1/9*Pi*3^(1/2), 3^(1/2), -(ln(abs(-1+2/3*3^(1/2)))+ln(3^(1/2))+1)/(ln
(abs(1+2/3*3^(1/2)))+ln(3^(1/2))+1), 8.3099863401554735233]




[x^2+3*x+1, 2/5*5^(1/2)*arccoth(3/5*5^(1/2))-2/5*5^(1/2)*arccoth(5^(1/2)), 5^(1
/2), -(ln(abs(1-2/5*5^(1/2)))+ln(5^(1/2))+1)/(ln(abs(-1-2/5*5^(1/2)))+ln(5^(1/2
))+1), 6.5082057497631049136]




[3*x^2+3*x+1, -2/9*Pi*3^(1/2)+2/3*arctan(3*3^(1/2))*3^(1/2), 3, -(ln(abs(-5/3+2
/3*7^(1/2)))+ln(3)+1)/(ln(abs(5/3+2/3*7^(1/2)))+ln(3)+1), 15.315550900340627733
]




[7*x^2+4*x+1, -1/3*arctan(2/3*3^(1/2))*3^(1/2)+1/3*arctan(3*3^(1/2))*3^(1/2), 2
*3^(1/2), -(ln(abs(-1/2+1/6*12^(1/2)))+ln(2*3^(1/2))+1)/(ln(abs(1/2+1/6*12^(1/2
)))+ln(2*3^(1/2))+1), 8.3099863401554735302]




[5*x^2+5*x+1, 2/5*5^(1/2)*arccoth(5^(1/2))-2/5*arccoth(3*5^(1/2))*5^(1/2), 5, -
(ln(abs(7/5-2/5*11^(1/2)))+ln(5)+1)/(ln(abs(-7/5-2/5*11^(1/2)))+ln(5)+1), 1176.\
2902524845007739]




[7*x^2+5*x+1, -2/3*arctan(5/3*3^(1/2))*3^(1/2)+2/3*arctan(19/3*3^(1/2))*3^(1/2)
, 3, -(ln(abs(-7/3+2/3*13^(1/2)))+ln(3)+1)/(ln(abs(7/3+2/3*13^(1/2)))+ln(3)+1),
7.5789513504094748015]




[8*x^2+5*x+1, -2/7*arctan(5/7*7^(1/2))*7^(1/2)+2/7*arctan(3*7^(1/2))*7^(1/2), 7
^(1/2), -(ln(abs(-1+2/7*14^(1/2)))+ln(7^(1/2))+1)/(ln(abs(1+2/7*14^(1/2)))+ln(7
^(1/2))+1), 4.8569705938289177946]




[10*x^2+6*x+1, -arctan(3)+arctan(13), 4, -(ln(abs(-2+1/2*17^(1/2)))+ln(4)+1)/(
ln(abs(2+1/2*17^(1/2)))+ln(4)+1), 10.432735096002347812]




[2*x^2+3*x+2, -2/7*arctan(3/7*7^(1/2))*7^(1/2)+2/7*arctan(7^(1/2))*7^(1/2), 7^(
1/2), -(ln(abs(-1+2/7*14^(1/2)))+ln(7^(1/2))+1)/(ln(abs(1+2/7*14^(1/2)))+ln(7^(
1/2))+1), 4.8569705938289177946]




[7*x^2+6*x+2, -1/5*arctan(3/5*5^(1/2))*5^(1/2)+1/5*arctan(2*5^(1/2))*5^(1/2), 2
*10^(1/2), -(ln(abs(-1/2+1/10*30^(1/2)))+ln(2*10^(1/2))+1)/(ln(abs(1/2+1/10*30^
(1/2)))+ln(2*10^(1/2))+1), 15.607845206093606153]




[7*x^2+7*x+2, -2/7*arctan(7^(1/2))*7^(1/2)+2/7*arctan(3*7^(1/2))*7^(1/2), 7, -(
ln(abs(-11/7+2/7*32^(1/2)))+ln(7)+1)/(ln(abs(11/7+2/7*32^(1/2)))+ln(7)+1), 26.7\
71696953576777408]




[7*x^2+8*x+2, 1/2*2^(1/2)*arccoth(2*2^(1/2))-1/2*2^(1/2)*arccoth(11/2*2^(1/2)),
4*2^(1/2), -(ln(abs(3/2-1/4*34^(1/2)))+ln(4*2^(1/2))+1)/(ln(abs(-3/2-1/4*34^(1/
2)))+ln(4*2^(1/2))+1), 9.8568612074158945874]




[9*x^2+8*x+2, -1/2*arctan(2*2^(1/2))*2^(1/2)+1/2*arctan(13/2*2^(1/2))*2^(1/2),
4*2^(1/2), -(ln(abs(-3/2+1/4*38^(1/2)))+ln(4*2^(1/2))+1)/(ln(abs(3/2+1/4*38^(1/
2)))+ln(4*2^(1/2))+1), 9.3808596317363403541]




[x^2+3*x+3, -2/9*Pi*3^(1/2)+2/3*arctan(5/3*3^(1/2))*3^(1/2), 3^(1/2), -(ln(abs(
-3+2/3*21^(1/2)))+ln(3^(1/2))+1)/(ln(abs(3+2/3*21^(1/2)))+ln(3^(1/2))+1), 3.481\
2632908955237101]




[3*x^2+4*x+3, -1/5*arctan(2/5*5^(1/2))*5^(1/2)+1/5*arctan(5^(1/2))*5^(1/2), 2*
10^(1/2), -(ln(abs(-1/2+1/10*30^(1/2)))+ln(2*10^(1/2))+1)/(ln(abs(1/2+1/10*30^(
1/2)))+ln(2*10^(1/2))+1), 15.607845206093606153]




[3*x^2+5*x+3, -2/11*arctan(5/11*11^(1/2))*11^(1/2)+2/11*arctan(11^(1/2))*11^(1/
2), 11^(1/2), -(ln(abs(-1+2/11*33^(1/2)))+ln(11^(1/2))+1)/(ln(abs(1+2/11*33^(1/
2)))+ln(11^(1/2))+1), 4.1879821595890442019]




[2*x^2+6*x+3, 1/3*arccoth(3^(1/2))*3^(1/2)-1/3*3^(1/2)*arccoth(5/3*3^(1/2)), 4*
3^(1/2), -(ln(abs(1-1/6*33^(1/2)))+ln(4*3^(1/2))+1)/(ln(abs(-1-1/6*33^(1/2)))+
ln(4*3^(1/2))+1), 17.326964322421861472]




[4*x^2+6*x+3, -1/9*Pi*3^(1/2)+1/3*arctan(7/3*3^(1/2))*3^(1/2), 4*3^(1/2), -(ln(
abs(-1+1/6*39^(1/2)))+ln(4*3^(1/2))+1)/(ln(abs(1+1/6*39^(1/2)))+ln(4*3^(1/2))+1
), 14.892104575824395585]




[7*x^2+6*x+3, -1/6*arctan(1/2*3^(1/2))*3^(1/2)+1/6*arctan(5/3*3^(1/2))*3^(1/2),
4*3^(1/2), -(ln(abs(-1/4+1/24*48^(1/2)))+ln(4*3^(1/2))+1)/(ln(abs(1/4+1/24*48^(
1/2)))+ln(4*3^(1/2))+1), 8.3099863401554735279]




[3*x^2+7*x+3, 2/13*arccoth(7/13*13^(1/2))*13^(1/2)-2/13*13^(1/2)*arccoth(13^(1/
2)), 13^(1/2), -(ln(abs(1-2/13*39^(1/2)))+ln(13^(1/2))+1)/(ln(abs(-1-2/13*39^(1
/2)))+ln(13^(1/2))+1), 4.0924630490843634450]




[6*x^2+8*x+3, -1/2*arctan(2*2^(1/2))*2^(1/2)+1/2*arctan(5*2^(1/2))*2^(1/2), 8,
-(ln(abs(-7/4+1/4*51^(1/2)))+ln(8)+1)/(ln(abs(7/4+1/4*51^(1/2)))+ln(8)+1), 17.5\
22143636180518454]




[7*x^2+9*x+3, -2/3*arctan(3*3^(1/2))*3^(1/2)+2/3*arctan(23/3*3^(1/2))*3^(1/2),
3^(1/2), -(ln(abs(-5+2/3*57^(1/2)))+ln(3^(1/2))+1)/(ln(abs(5+2/3*57^(1/2)))+ln(
3^(1/2))+1), 3.0780462199168187258]




[8*x^2+9*x+3, -2/15*arctan(3/5*15^(1/2))*15^(1/2)+2/15*arctan(5/3*15^(1/2))*15^
(1/2), 15^(1/2), -(ln(abs(-1+2/15*60^(1/2)))+ln(15^(1/2))+1)/(ln(abs(1+2/15*60^
(1/2)))+ln(15^(1/2))+1), 3.8806942681404861577]




[x^2+2*x+4, -1/18*Pi*3^(1/2)+1/3*arctan(2/3*3^(1/2))*3^(1/2), 6, -(ln(abs(-5/6+
1/6*28^(1/2)))+ln(6)+1)/(ln(abs(5/6+1/6*28^(1/2)))+ln(6)+1), 15.315550900340627\
783]




[x^2+3*x+4, -2/7*arctan(3/7*7^(1/2))*7^(1/2)+2/7*arctan(5/7*7^(1/2))*7^(1/2), 7
, -(ln(abs(-11/7+2/7*32^(1/2)))+ln(7)+1)/(ln(abs(11/7+2/7*32^(1/2)))+ln(7)+1),
26.771696953576777408]




[2*x^2+5*x+4, -2/7*arctan(5/7*7^(1/2))*7^(1/2)+2/7*arctan(9/7*7^(1/2))*7^(1/2),
7, -(ln(abs(-13/7+2/7*44^(1/2)))+ln(7)+1)/(ln(abs(13/7+2/7*44^(1/2)))+ln(7)+1),
14.239775923153759269]




[x^2+6*x+4, 1/5*5^(1/2)*arccoth(3/5*5^(1/2))-1/5*5^(1/2)*arccoth(4/5*5^(1/2)),
10, -(ln(abs(7/10-1/10*44^(1/2)))+ln(10)+1)/(ln(abs(-7/10-1/10*44^(1/2)))+ln(10
)+1), 1176.2902524845005445]




[3*x^2+6*x+4, -1/9*Pi*3^(1/2)+1/3*arctan(2*3^(1/2))*3^(1/2), 6, -(ln(abs(-7/6+1
/6*52^(1/2)))+ln(6)+1)/(ln(abs(7/6+1/6*52^(1/2)))+ln(6)+1), 7.57895135040947478\
47]




[4*x^2+7*x+4, -2/15*arctan(7/15*15^(1/2))*15^(1/2)+2/15*arctan(15^(1/2))*15^(1/
2), 15^(1/2), -(ln(abs(-1+2/15*60^(1/2)))+ln(15^(1/2))+1)/(ln(abs(1+2/15*60^(1/
2)))+ln(15^(1/2))+1), 3.8806942681404861577]




[5*x^2+8*x+4, -1/2*arctan(2)+1/2*arctan(9/2), 8, -(ln(abs(-1+1/8*68^(1/2)))+ln(
8)+1)/(ln(abs(1+1/8*68^(1/2)))+ln(8)+1), 10.432735096002347775]




[4*x^2+9*x+4, 2/17*arccoth(9/17*17^(1/2))*17^(1/2)-2/17*17^(1/2)*arccoth(17^(1/
2)), 17^(1/2), -(ln(abs(1-2/17*68^(1/2)))+ln(17^(1/2))+1)/(ln(abs(-1-2/17*68^(1
/2)))+ln(17^(1/2))+1), 3.8269636216410127965]




[5*x^2+10*x+4, 1/5*5^(1/2)*arccoth(5^(1/2))-1/5*5^(1/2)*arccoth(2*5^(1/2)), 10,
-(ln(abs(9/10-1/10*76^(1/2)))+ln(10)+1)/(ln(abs(-9/10-1/10*76^(1/2)))+ln(10)+1)
, 15.613751960540034505]




[7*x^2+10*x+4, -1/3*arctan(5/3*3^(1/2))*3^(1/2)+1/3*arctan(4*3^(1/2))*3^(1/2),
2*3^(1/2), -(ln(abs(-3/2+1/6*84^(1/2)))+ln(2*3^(1/2))+1)/(ln(abs(3/2+1/6*84^(1/
2)))+ln(2*3^(1/2))+1), 3.4812632908955237133]




[x^2+2*x+5, -1/2*arctan(1/2)+1/8*Pi, 4*2^(1/2), -(ln(abs(-3/4+1/8*40^(1/2)))+ln
(4*2^(1/2))+1)/(ln(abs(3/4+1/8*40^(1/2)))+ln(4*2^(1/2))+1), 7.70735754852937098\
34]




[x^2+5*x+5, 2/5*5^(1/2)*arccoth(5^(1/2))-2/5*5^(1/2)*arccoth(7/5*5^(1/2)), 5^(1
/2), -(ln(abs(3-2/5*55^(1/2)))+ln(5^(1/2))+1)/(ln(abs(-3-2/5*55^(1/2)))+ln(5^(1
/2))+1), 3.2571690113448376880]




[2*x^2+5*x+5, -2/15*arctan(1/3*15^(1/2))*15^(1/2)+2/15*arctan(3/5*15^(1/2))*15^
(1/2), 15^(1/2), -(ln(abs(-1+2/15*60^(1/2)))+ln(15^(1/2))+1)/(ln(abs(1+2/15*60^
(1/2)))+ln(15^(1/2))+1), 3.8806942681404861577]




[2*x^2+6*x+5, -arctan(3)+arctan(5), 4, -(ln(abs(-4+1/2*65^(1/2)))+ln(4)+1)/(ln(
abs(4+1/2*65^(1/2)))+ln(4)+1), 5.1258339353606460401]




[3*x^2+7*x+5, -2/11*arctan(7/11*11^(1/2))*11^(1/2)+2/11*arctan(13/11*11^(1/2))*
11^(1/2), 11, -(ln(abs(-17/11+2/11*75^(1/2)))+ln(11)+1)/(ln(abs(17/11+2/11*75^(
1/2)))+ln(11)+1), 33.903991854273836881]




[5*x^2+7*x+5, -2/51*arctan(7/51*51^(1/2))*51^(1/2)+2/51*arctan(1/3*51^(1/2))*51
^(1/2), 3*17^(1/2), -(ln(abs(-1/3+2/51*85^(1/2)))+ln(3*17^(1/2))+1)/(ln(abs(1/3
+2/51*85^(1/2)))+ln(3*17^(1/2))+1), 60.912947227427311286]




[5*x^2+9*x+5, -2/19*arctan(9/19*19^(1/2))*19^(1/2)+2/19*arctan(19^(1/2))*19^(1/
2), 19^(1/2), -(ln(abs(-1+2/19*95^(1/2)))+ln(19^(1/2))+1)/(ln(abs(1+2/19*95^(1/
2)))+ln(19^(1/2))+1), 3.6974006367249319472]




[x^2+10*x+5, 1/10*arccoth(1/2*5^(1/2))*5^(1/2)-1/10*5^(1/2)*arccoth(3/5*5^(1/2)
), 4*5^(1/2), -(ln(abs(1/4-1/40*80^(1/2)))+ln(4*5^(1/2))+1)/(ln(abs(-1/4-1/40*
80^(1/2)))+ln(4*5^(1/2))+1), 6.5082057497631049136]




[3*x^2+10*x+5, 1/10*10^(1/2)*arccoth(1/2*10^(1/2))-1/10*10^(1/2)*arccoth(4/5*10
^(1/2)), 4*10^(1/2), -(ln(abs(1/2-1/20*90^(1/2)))+ln(4*10^(1/2))+1)/(ln(abs(-1/
2-1/20*90^(1/2)))+ln(4*10^(1/2))+1), 29.025649285508016267]




[4*x^2+10*x+5, 1/5*5^(1/2)*arccoth(5^(1/2))-1/5*arccoth(9/5*5^(1/2))*5^(1/2), 4
*5^(1/2), -(ln(abs(1-1/10*95^(1/2)))+ln(4*5^(1/2))+1)/(ln(abs(-1-1/10*95^(1/2))
)+ln(4*5^(1/2))+1), 8.9802507673707511949]




[6*x^2+10*x+5, -1/5*arctan(5^(1/2))*5^(1/2)+1/5*arctan(11/5*5^(1/2))*5^(1/2), 4
*5^(1/2), -(ln(abs(-1+1/10*105^(1/2)))+ln(4*5^(1/2))+1)/(ln(abs(1+1/10*105^(1/2
)))+ln(4*5^(1/2))+1), 8.6379958426045502068]




[7*x^2+10*x+5, -1/10*arctan(1/2*10^(1/2))*10^(1/2)+1/10*arctan(6/5*10^(1/2))*10
^(1/2), 4*10^(1/2), -(ln(abs(-1/2+1/20*110^(1/2)))+ln(4*10^(1/2))+1)/(ln(abs(1/
2+1/20*110^(1/2)))+ln(4*10^(1/2))+1), 21.305691483920764832]




[9*x^2+10*x+5, -1/10*arctan(1/2*5^(1/2))*5^(1/2)+1/10*arctan(7/5*5^(1/2))*5^(1/
2), 4*10^(1/2), -(ln(abs(-1/4+1/40*120^(1/2)))+ln(4*10^(1/2))+1)/(ln(abs(1/4+1/
40*120^(1/2)))+ln(4*10^(1/2))+1), 15.607845206093606123]




[x^2+3*x+6, -2/15*arctan(1/5*15^(1/2))*15^(1/2)+2/15*arctan(1/3*15^(1/2))*15^(1
/2), 15^(1/2), -(ln(abs(-1+2/15*60^(1/2)))+ln(15^(1/2))+1)/(ln(abs(1+2/15*60^(1
/2)))+ln(15^(1/2))+1), 3.8806942681404861577]




[x^2+6*x+6, 1/3*arccoth(3^(1/2))*3^(1/2)-1/3*arccoth(4/3*3^(1/2))*3^(1/2), 2*6^
(1/2), -(ln(abs(3/2-1/6*78^(1/2)))+ln(2*6^(1/2))+1)/(ln(abs(-3/2-1/6*78^(1/2)))
+ln(2*6^(1/2))+1), 4.7338779958071950700]




[x^2+8*x+6, 1/10*10^(1/2)*arccoth(2/5*10^(1/2))-1/10*10^(1/2)*arccoth(1/2*10^(1
/2)), 4*10^(1/2), -(ln(abs(1/2-1/20*90^(1/2)))+ln(4*10^(1/2))+1)/(ln(abs(-1/2-1
/20*90^(1/2)))+ln(4*10^(1/2))+1), 29.025649285508016267]




[3*x^2+8*x+6, -1/2*arctan(2*2^(1/2))*2^(1/2)+1/2*arctan(7/2*2^(1/2))*2^(1/2), 4
*2^(1/2), -(ln(abs(-5/2+1/4*102^(1/2)))+ln(4*2^(1/2))+1)/(ln(abs(5/2+1/4*102^(1
/2)))+ln(4*2^(1/2))+1), 5.5238132377130025906]




[4*x^2+9*x+6, -2/15*arctan(3/5*15^(1/2))*15^(1/2)+2/15*arctan(17/15*15^(1/2))*
15^(1/2), 5*3^(1/2), -(ln(abs(-7/5+2/15*114^(1/2)))+ln(5*3^(1/2))+1)/(ln(abs(7/
5+2/15*114^(1/2)))+ln(5*3^(1/2))+1), 8.1455807933567691820]




[5*x^2+10*x+6, -1/5*arctan(5^(1/2))*5^(1/2)+1/5*arctan(2*5^(1/2))*5^(1/2), 10*2
^(1/2), -(ln(abs(-11/10+1/10*126^(1/2)))+ln(10*2^(1/2))+1)/(ln(abs(11/10+1/10*
126^(1/2)))+ln(10*2^(1/2))+1), 31.631103067531731805]




[x^2+x+7, -2/9*arctan(1/9*3^(1/2))*3^(1/2)+1/27*Pi*3^(1/2), 9, -(ln(abs(-5/9+2/
27*63^(1/2)))+ln(9)+1)/(ln(abs(5/9+2/27*63^(1/2)))+ln(9)+1), 15.315550900340627\
740]




[x^2+4*x+7, -1/3*arctan(2/3*3^(1/2))*3^(1/2)+1/9*Pi*3^(1/2), 2*3^(1/2), -(ln(
abs(-3/2+1/6*84^(1/2)))+ln(2*3^(1/2))+1)/(ln(abs(3/2+1/6*84^(1/2)))+ln(2*3^(1/2
))+1), 3.4812632908955237133]




[x^2+5*x+7, -2/3*arctan(5/3*3^(1/2))*3^(1/2)+2/3*arctan(7/3*3^(1/2))*3^(1/2), 3
, -(ln(abs(-19/3+2/3*91^(1/2)))+ln(3)+1)/(ln(abs(19/3+2/3*91^(1/2)))+ln(3)+1),
4.0107214983056850938]




[x^2+6*x+7, 1/2*2^(1/2)*arccoth(3/2*2^(1/2))-1/2*2^(1/2)*arccoth(2*2^(1/2)), 4*
2^(1/2), -(ln(abs(5/2-1/4*98^(1/2)))+ln(4*2^(1/2))+1)/(ln(abs(-5/2-1/4*98^(1/2)
))+ln(4*2^(1/2))+1), 5.5608711225648792435]




[2*x^2+6*x+7, -1/5*arctan(3/5*5^(1/2))*5^(1/2)+1/5*arctan(5^(1/2))*5^(1/2), 4*5
^(1/2), -(ln(abs(-1+1/10*105^(1/2)))+ln(4*5^(1/2))+1)/(ln(abs(1+1/10*105^(1/2))
)+ln(4*5^(1/2))+1), 8.6379958426045502068]




[3*x^2+6*x+7, -1/6*arctan(1/2*3^(1/2))*3^(1/2)+1/18*Pi*3^(1/2), 12, -(ln(abs(-5
/12+1/24*112^(1/2)))+ln(12)+1)/(ln(abs(5/12+1/24*112^(1/2)))+ln(12)+1), 15.3155\
50900340627746]




[x^2+7*x+7, 2/21*21^(1/2)*arccoth(1/3*21^(1/2))-2/21*21^(1/2)*arccoth(3/7*21^(1
/2)), 21^(1/2), -(ln(abs(1-2/21*105^(1/2)))+ln(21^(1/2))+1)/(ln(abs(-1-2/21*105
^(1/2)))+ln(21^(1/2))+1), 3.6621162818653647041]




[2*x^2+7*x+7, -2/7*arctan(7^(1/2))*7^(1/2)+2/7*arctan(11/7*7^(1/2))*7^(1/2), 7^
(1/2), -(ln(abs(-3+2/7*112^(1/2)))+ln(7^(1/2))+1)/(ln(abs(3+2/7*112^(1/2)))+ln(
7^(1/2))+1), 3.1307999826699309253]




[3*x^2+7*x+7, -2/35*arctan(1/5*35^(1/2))*35^(1/2)+2/35*arctan(13/35*35^(1/2))*
35^(1/2), 5*7^(1/2), -(ln(abs(-3/5+2/35*119^(1/2)))+ln(5*7^(1/2))+1)/(ln(abs(3/
5+2/35*119^(1/2)))+ln(5*7^(1/2))+1), 22.678279240725504417]




[4*x^2+7*x+7, -2/21*arctan(1/3*7^(1/2))*7^(1/2)+2/21*arctan(5/7*7^(1/2))*7^(1/2
), 3*7^(1/2), -(ln(abs(-1/3+2/63*126^(1/2)))+ln(3*7^(1/2))+1)/(ln(abs(1/3+2/63*
126^(1/2)))+ln(3*7^(1/2))+1), 4.8569705938289177923]




[2*x^2+8*x+7, 1/2*2^(1/2)*arccoth(2*2^(1/2))-1/2*arccoth(3*2^(1/2))*2^(1/2), 8,
-(ln(abs(11/4-1/4*119^(1/2)))+ln(8)+1)/(ln(abs(-11/4-1/4*119^(1/2)))+ln(8)+1),
7.8228657034259635161]




[3*x^2+8*x+7, -1/5*arctan(4/5*5^(1/2))*5^(1/2)+1/5*arctan(7/5*5^(1/2))*5^(1/2),
10*2^(1/2), -(ln(abs(-11/10+1/10*126^(1/2)))+ln(10*2^(1/2))+1)/(ln(abs(11/10+1/
10*126^(1/2)))+ln(10*2^(1/2))+1), 31.631103067531731805]




[3*x^2+9*x+7, -2/3*arctan(3*3^(1/2))*3^(1/2)+2/3*arctan(5*3^(1/2))*3^(1/2), 3,
-(ln(abs(-23/3+2/3*133^(1/2)))+ln(3)+1)/(ln(abs(23/3+2/3*133^(1/2)))+ln(3)+1),
3.7896117451863288465]




[4*x^2+10*x+7, -1/3*arctan(5/3*3^(1/2))*3^(1/2)+1/3*arctan(3*3^(1/2))*3^(1/2),
4*3^(1/2), -(ln(abs(-2+1/6*147^(1/2)))+ln(4*3^(1/2))+1)/(ln(abs(2+1/6*147^(1/2)
))+ln(4*3^(1/2))+1), 5.5994975951404949745]




[x^2+4*x+8, -1/8*Pi+1/2*arctan(3/2), 4*2^(1/2), -(ln(abs(-5/4+1/8*104^(1/2)))+
ln(4*2^(1/2))+1)/(ln(abs(5/4+1/8*104^(1/2)))+ln(4*2^(1/2))+1), 4.78832743175661\
98544]




[3*x^2+4*x+8, -1/10*arctan(1/5*5^(1/2))*5^(1/2)+1/10*arctan(1/2*5^(1/2))*5^(1/2
), 4*10^(1/2), -(ln(abs(-1/4+1/40*120^(1/2)))+ln(4*10^(1/2))+1)/(ln(abs(1/4+1/
40*120^(1/2)))+ln(4*10^(1/2))+1), 15.607845206093606123]




[x^2+5*x+8, -2/7*arctan(5/7*7^(1/2))*7^(1/2)+2/7*arctan(7^(1/2))*7^(1/2), 7^(1/
2), -(ln(abs(-3+2/7*112^(1/2)))+ln(7^(1/2))+1)/(ln(abs(3+2/7*112^(1/2)))+ln(7^(
1/2))+1), 3.1307999826699309253]




[2*x^2+7*x+8, -2/15*arctan(7/15*15^(1/2))*15^(1/2)+2/15*arctan(11/15*15^(1/2))*
15^(1/2), 15, -(ln(abs(-23/15+2/15*136^(1/2)))+ln(15)+1)/(ln(abs(23/15+2/15*136
^(1/2)))+ln(15)+1), 38.895259299026773802]




[x^2+8*x+8, 1/4*2^(1/2)*arccoth(2^(1/2))-1/4*arccoth(5/4*2^(1/2))*2^(1/2), 8*2^
(1/2), -(ln(abs(3/4-1/16*136^(1/2)))+ln(8*2^(1/2))+1)/(ln(abs(-3/4-1/16*136^(1/
2)))+ln(8*2^(1/2))+1), 9.8568612074158945565]




[3*x^2+8*x+8, -1/4*arctan(2^(1/2))*2^(1/2)+1/4*arctan(7/4*2^(1/2))*2^(1/2), 8*2
^(1/2), -(ln(abs(-3/4+1/16*152^(1/2)))+ln(8*2^(1/2))+1)/(ln(abs(3/4+1/16*152^(1
/2)))+ln(8*2^(1/2))+1), 9.3808596317363403614]




[2*x^2+9*x+8, 2/17*arccoth(9/17*17^(1/2))*17^(1/2)-2/17*17^(1/2)*arccoth(13/17*
17^(1/2)), 17, -(ln(abs(25/17-2/17*152^(1/2)))+ln(17)+1)/(ln(abs(-25/17-2/17*
152^(1/2)))+ln(17)+1), 69.184664932665641876]




[3*x^2+9*x+8, -2/15*arctan(3/5*15^(1/2))*15^(1/2)+2/15*arctan(15^(1/2))*15^(1/2
), 3*5^(1/2), -(ln(abs(-5/3+2/15*160^(1/2)))+ln(3*5^(1/2))+1)/(ln(abs(5/3+2/15*
160^(1/2)))+ln(3*5^(1/2))+1), 5.0539077534823343831]




[x^2+3*x+9, -1/27*Pi*3^(1/2)+2/9*arctan(5/9*3^(1/2))*3^(1/2), 9, -(ln(abs(-7/9+
2/27*117^(1/2)))+ln(9)+1)/(ln(abs(7/9+2/27*117^(1/2)))+ln(9)+1), 7.578951350409\
4747810]




[2*x^2+3*x+9, -2/21*arctan(1/7*7^(1/2))*7^(1/2)+2/21*arctan(1/3*7^(1/2))*7^(1/2
), 3*7^(1/2), -(ln(abs(-1/3+2/63*126^(1/2)))+ln(3*7^(1/2))+1)/(ln(abs(1/3+2/63*
126^(1/2)))+ln(3*7^(1/2))+1), 4.8569705938289177923]




[x^2+4*x+9, -1/5*arctan(2/5*5^(1/2))*5^(1/2)+1/5*arctan(3/5*5^(1/2))*5^(1/2), 
10*2^(1/2), -(ln(abs(-11/10+1/10*126^(1/2)))+ln(10*2^(1/2))+1)/(ln(abs(11/10+1/
10*126^(1/2)))+ln(10*2^(1/2))+1), 31.631103067531731805]




[x^2+5*x+9, -2/11*arctan(5/11*11^(1/2))*11^(1/2)+2/11*arctan(7/11*11^(1/2))*11^
(1/2), 11, -(ln(abs(-23/11+2/11*135^(1/2)))+ln(11)+1)/(ln(abs(23/11+2/11*135^(1
/2)))+ln(11)+1), 12.089124796145223893]




[2*x^2+6*x+9, -1/12*Pi+1/3*arctan(5/3), 12, -(ln(abs(-2/3+1/18*153^(1/2)))+ln(
12)+1)/(ln(abs(2/3+1/18*153^(1/2)))+ln(12)+1), 10.432735096002347784]




[x^2+7*x+9, 2/13*arccoth(7/13*13^(1/2))*13^(1/2)-2/13*arccoth(9/13*13^(1/2))*13
^(1/2), 13, -(ln(abs(25/13-2/13*153^(1/2)))+ln(13)+1)/(ln(abs(-25/13-2/13*153^(
1/2)))+ln(13)+1), 15.354341238420183413]




[2*x^2+8*x+9, -1/2*arctan(2*2^(1/2))*2^(1/2)+1/2*arctan(3*2^(1/2))*2^(1/2), 8,
-(ln(abs(-13/4+1/4*171^(1/2)))+ln(8)+1)/(ln(abs(13/4+1/4*171^(1/2)))+ln(8)+1),
6.6635635425939131729]




[x^2+9*x+9, 2/15*5^(1/2)*arccoth(3/5*5^(1/2))-2/15*arccoth(11/15*5^(1/2))*5^(1/
2), 15, -(ln(abs(3/5-2/45*171^(1/2)))+ln(15)+1)/(ln(abs(-3/5-2/45*171^(1/2)))+
ln(15)+1), 15.613751960540034478]




[2*x^2+10*x+9, 1/7*arccoth(5/7*7^(1/2))*7^(1/2)-1/7*7^(1/2)*arccoth(7^(1/2)), 4
*7^(1/2), -(ln(abs(1-1/14*189^(1/2)))+ln(4*7^(1/2))+1)/(ln(abs(-1-1/14*189^(1/2
)))+ln(4*7^(1/2))+1), 7.1537717511215774039]




[3*x^2+10*x+9, -1/2*arctan(5/2*2^(1/2))*2^(1/2)+1/2*arctan(4*2^(1/2))*2^(1/2),
4*2^(1/2), -(ln(abs(-7/2+1/4*198^(1/2)))+ln(4*2^(1/2))+1)/(ln(abs(7/2+1/4*198^(
1/2)))+ln(4*2^(1/2))+1), 4.6148485335234683752]




[x^2+5*x+10, -2/15*arctan(1/3*15^(1/2))*15^(1/2)+2/15*arctan(7/15*15^(1/2))*15^
(1/2), 3*5^(1/2), -(ln(abs(-5/3+2/15*160^(1/2)))+ln(3*5^(1/2))+1)/(ln(abs(5/3+2
/15*160^(1/2)))+ln(3*5^(1/2))+1), 5.0539077534823343831]




[x^2+8*x+10, 1/6*arccoth(2/3*6^(1/2))*6^(1/2)-1/6*arccoth(5/6*6^(1/2))*6^(1/2),
12*2^(1/2), -(ln(abs(7/6-1/12*190^(1/2)))+ln(12*2^(1/2))+1)/(ln(abs(-7/6-1/12*
190^(1/2)))+ln(12*2^(1/2))+1), 26.095688442030019397]




[3*x^2+8*x+10, -1/14*arctan(2/7*14^(1/2))*14^(1/2)+1/14*arctan(1/2*14^(1/2))*14
^(1/2), 4*14^(1/2), -(ln(abs(-1/2+1/28*210^(1/2)))+ln(4*14^(1/2))+1)/(ln(abs(1/
2+1/28*210^(1/2)))+ln(4*14^(1/2))+1), 12.050569329465964557]




[3*x^2+10*x+10, -1/5*arctan(5^(1/2))*5^(1/2)+1/5*arctan(8/5*5^(1/2))*5^(1/2), 2
*10^(1/2), -(ln(abs(-3/2+1/10*230^(1/2)))+ln(2*10^(1/2))+1)/(ln(abs(3/2+1/10*
230^(1/2)))+ln(2*10^(1/2))+1), 4.1452252161237347194]




[7*x^2+10*x+10, -1/15*arctan(1/3*5^(1/2))*5^(1/2)+1/15*arctan(4/5*5^(1/2))*5^(1
/2), 6*10^(1/2), -(ln(abs(-1/6+1/90*270^(1/2)))+ln(6*10^(1/2))+1)/(ln(abs(1/6+1
/90*270^(1/2)))+ln(6*10^(1/2))+1), 15.607845206093606123]




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