\(\int \frac {A+B x+C x^2}{(70+67 x-53 x^2+6 x^3)^{5/2}} \, dx\) [122]

Optimal result

Integrand size = 30, antiderivative size = 475 \[ \int \frac {A+B x+C x^2}{\left (70+67 x-53 x^2+6 x^3\right )^{5/2}} \, dx=-\frac {2 (A+7 B+49 C) (5-2 x) (7-x) (2+3 x)}{621 \left (70+67 x-53 x^2+6 x^3\right )^{5/2}}-\frac {2 (146 A+815 B+4256 C) (5-2 x) (7-x)^2 (2+3 x)}{42849 \left (70+67 x-53 x^2+6 x^3\right )^{5/2}}+\frac {4 (11312 A+57863 B+298643 C) (5-2 x) (7-x)^3 (2+3 x)}{21981537 \left (70+67 x-53 x^2+6 x^3\right )^{5/2}}-\frac {4 (170170 A+1604983 B+8818156 C) (5-2 x)^2 (7-x)^3 (2+3 x)}{3758842827 \left (70+67 x-53 x^2+6 x^3\right )^{5/2}}-\frac {10 (62482 A-7760240 B-45931763 C) (5-2 x)^3 (7-x)^3 (2+3 x)}{547538105133 \left (70+67 x-53 x^2+6 x^3\right )^{5/2}}-\frac {4 (243399520 A-417945287 B-3367220459 C) (5-2 x)^3 (7-x)^3 (2+3 x)^2}{239274151943121 \left (70+67 x-53 x^2+6 x^3\right )^{5/2}}-\frac {4 (243399520 A-417945287 B-3367220459 C) (5-2 x)^{5/2} (7-x)^{5/2} (2+3 x)^{5/2} E\left (\arcsin \left (\frac {\sqrt {2+3 x}}{\sqrt {23}}\right )|\frac {46}{19}\right )}{37780129254177 \sqrt {19} \left (70+67 x-53 x^2+6 x^3\right )^{5/2}}-\frac {4 (8748670 A+17573419 B+70923418 C) (-7+x)^{5/2} (-5+2 x)^{5/2} (2+3 x)^{5/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {19}{2}}}{\sqrt {2+3 x}}\right ),\frac {46}{19}\right )}{1399264046451 \sqrt {19} \left (70+67 x-53 x^2+6 x^3\right )^{5/2}} \] Output:

-2/621*(A+7*B+49*C)*(5-2*x)*(7-x)*(2+3*x)/(6*x^3-53*x^2+67*x+70)^(5/2)-2/4 
2849*(146*A+815*B+4256*C)*(5-2*x)*(7-x)^2*(2+3*x)/(6*x^3-53*x^2+67*x+70)^( 
5/2)+4/21981537*(11312*A+57863*B+298643*C)*(5-2*x)*(7-x)^3*(2+3*x)/(6*x^3- 
53*x^2+67*x+70)^(5/2)-4/3758842827*(170170*A+1604983*B+8818156*C)*(5-2*x)^ 
2*(7-x)^3*(2+3*x)/(6*x^3-53*x^2+67*x+70)^(5/2)-10/547538105133*(62482*A-77 
60240*B-45931763*C)*(5-2*x)^3*(7-x)^3*(2+3*x)/(6*x^3-53*x^2+67*x+70)^(5/2) 
-4/239274151943121*(243399520*A-417945287*B-3367220459*C)*(5-2*x)^3*(7-x)^ 
3*(2+3*x)^2/(6*x^3-53*x^2+67*x+70)^(5/2)-4/717822455829363*(243399520*A-41 
7945287*B-3367220459*C)*(5-2*x)^(5/2)*(7-x)^(5/2)*(2+3*x)^(5/2)*EllipticE( 
1/23*(2+3*x)^(1/2)*23^(1/2),1/19*874^(1/2))*19^(1/2)/(6*x^3-53*x^2+67*x+70 
)^(5/2)-4/26586016882569*(8748670*A[phone]*B[phone]*C)*(-7+x)^(5/2)*(- 
5+2*x)^(5/2)*(2+3*x)^(5/2)*EllipticF(1/2*38^(1/2)/(2+3*x)^(1/2),1/19*874^( 
1/2))*19^(1/2)/(6*x^3-53*x^2+67*x+70)^(5/2)
 

Mathematica [A] (verified)

Time = 10.52 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.49 \[ \int \frac {A+B x+C x^2}{\left (70+67 x-53 x^2+6 x^3\right )^{5/2}} \, dx=-\frac {2 \left (15468489 \left (A \left (13687-125932 x+19236 x^2\right )+B \left (-224420-201115 x+43986 x^2\right )+C \left (-513170-715597 x+187428 x^2\right )\right )+\left (70+67 x-53 x^2+6 x^3\right ) \left (C \left (210892425319+183141328016 x-40406645508 x^2\right )+B \left (52671353512+13733513918 x-5015343444 x^2\right )+70 A \left (381044737-392492842 x+41725632 x^2\right )\right )+\sqrt {46} \sqrt {5-2 x} \sqrt {7-x} \sqrt {2+3 x} \left (70+67 x-53 x^2+6 x^3\right ) \left ((486799040 A-835890574 B-6734440918 C) E\left (\arcsin \left (\sqrt {\frac {2}{19}} \sqrt {2+3 x}\right )|\frac {19}{46}\right )+27 (-17809750 A+3654362 B+87811979 C) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{19}} \sqrt {2+3 x}\right ),\frac {19}{46}\right )\right )\right )}{717822455829363 \left (70+67 x-53 x^2+6 x^3\right )^{3/2}} \] Input:

Integrate[(A + B*x + C*x^2)/(70 + 67*x - 53*x^2 + 6*x^3)^(5/2),x]
 

Output:

(-2*(15468489*(A*(13687 - 125932*x + 19236*x^2) + B*(-224420 - 201115*x + 
43986*x^2) + C*(-513170 - 715597*x + 187428*x^2)) + (70 + 67*x - 53*x^2 + 
6*x^3)*(C*(210892425319 + 183141328016*x - 40406645508*x^2) + B*(526713535 
12 + 13733513918*x - 5015343444*x^2) + 70*A*(381044737 - 392492842*x + 417 
25632*x^2)) + Sqrt[46]*Sqrt[5 - 2*x]*Sqrt[7 - x]*Sqrt[2 + 3*x]*(70 + 67*x 
- 53*x^2 + 6*x^3)*((486799040*A - 835890574*B - 6734440918*C)*EllipticE[Ar 
cSin[Sqrt[2/19]*Sqrt[2 + 3*x]], 19/46] + 27*(-17809750*A + 3654362*B + 878 
11979*C)*EllipticF[ArcSin[Sqrt[2/19]*Sqrt[2 + 3*x]], 19/46])))/(7178224558 
29363*(70 + 67*x - 53*x^2 + 6*x^3)^(3/2))
 

Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 19.56 (sec) , antiderivative size = 2899, normalized size of antiderivative = 6.10, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {2526, 2490, 2486, 27, 1235, 27, 1235, 27, 1237, 27, 1237, 27, 1269, 1172, 321, 327}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {A+B x+C x^2}{\left (6 x^3-53 x^2+67 x+70\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 2526

\(\displaystyle \frac {1}{18} \int \frac {18 A-67 C+2 (9 B+53 C) x}{\left (6 x^3-53 x^2+67 x+70\right )^{5/2}}dx-\frac {C}{27 \left (6 x^3-53 x^2+67 x+70\right )^{3/2}}\)

\(\Big \downarrow \) 2490

\(\displaystyle \frac {1}{18} \int \frac {\frac {1}{18} (18 (18 A-67 C)+106 (9 B+53 C))+2 (9 B+53 C) \left (x-\frac {53}{18}\right )}{\left (6 \left (x-\frac {53}{18}\right )^3-\frac {1603}{18} \left (x-\frac {53}{18}\right )-\frac {9490}{243}\right )^{5/2}}d\left (x-\frac {53}{18}\right )-\frac {C}{27 \left (6 x^3-53 x^2+67 x+70\right )^{3/2}}\)

\(\Big \downarrow \) 2486

\(\displaystyle -\frac {C}{27 \left (6 x^3-53 x^2+67 x+70\right )^{3/2}}+\frac {54 \sqrt {2} \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{5/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{5/2} \int \frac {243 \sqrt {3} \left (162 A+477 B+2206 C+18 (9 B+53 C) \left (x-\frac {53}{18}\right )\right )}{\left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{5/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{5/2}}d\left (x-\frac {53}{18}\right )}{\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {C}{27 \left (6 x^3-53 x^2+67 x+70\right )^{3/2}}+\frac {13122 \sqrt {6} \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{5/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{5/2} \int \frac {162 A+477 B+2206 C+18 (9 B+53 C) \left (x-\frac {53}{18}\right )}{\left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{5/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{5/2}}d\left (x-\frac {53}{18}\right )}{\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{5/2}}\)

\(\Big \downarrow \) 1235

\(\displaystyle \frac {13122 \sqrt {6} \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{5/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{5/2} \left (\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \int \frac {157464 \left (\frac {\left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (5139218-14427 \left (18980+35397 i \sqrt {3}\right )^{2/3}+2 \left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) (9 B+53 C)+7 \sqrt [3]{18980+35397 i \sqrt {3}} \left (2569609-916 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) (162 A+477 B+2206 C)}{18980+35397 i \sqrt {3}}-\frac {126 \left (i \left (2569609 i+\left (18980 i-35397 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}\right ) (9 B+53 C)+\sqrt [3]{18980+35397 i \sqrt {3}} \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (162 A+477 B+2206 C)\right ) \left (x-\frac {53}{18}\right )}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )}{\left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{5/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{3/2}}d\left (x-\frac {53}{18}\right )}{100327557744 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}-\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (9 (3206 A+7331 B+31238 C)-\frac {\left (\left (2569609+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) (9 B+53 C)-\sqrt [3]{18980+35397 i \sqrt {3}} \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (162 A+477 B+2206 C)\right ) \left (x-\frac {53}{18}\right )}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )}{318573 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{3/2}}\right )}{\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{5/2}}-\frac {C}{27 \left (6 x^3-53 x^2+67 x+70\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {13122 \sqrt {6} \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{5/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{5/2} \left (\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \int \frac {\frac {\left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (5139218-14427 \left (18980+35397 i \sqrt {3}\right )^{2/3}+2 \left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) (9 B+53 C)+7 \sqrt [3]{18980+35397 i \sqrt {3}} \left (2569609-916 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) (162 A+477 B+2206 C)}{18980+35397 i \sqrt {3}}-\frac {126 \left (i \left (2569609 i+\left (18980 i-35397 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}\right ) (9 B+53 C)+\sqrt [3]{18980+35397 i \sqrt {3}} \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (162 A+477 B+2206 C)\right ) \left (x-\frac {53}{18}\right )}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}}{\left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{5/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{3/2}}d\left (x-\frac {53}{18}\right )}{637146 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}-\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (9 (3206 A+7331 B+31238 C)-\frac {\left (\left (2569609+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) (9 B+53 C)-\sqrt [3]{18980+35397 i \sqrt {3}} \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (162 A+477 B+2206 C)\right ) \left (x-\frac {53}{18}\right )}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )}{318573 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{3/2}}\right )}{\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{5/2}}-\frac {C}{27 \left (6 x^3-53 x^2+67 x+70\right )^{3/2}}\)

\(\Big \downarrow \) 1235

\(\displaystyle \frac {13122 \sqrt {6} \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{5/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{5/2} \left (\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (\frac {i \left (\frac {7 i \left (2 \left (13674725584000 i-171306047863719 \sqrt {3}-1468348 \left (18980 i-35397 \sqrt {3}\right ) \left (18980+35397 i \sqrt {3}\right )^{2/3}-1603 \sqrt [3]{18980+35397 i \sqrt {3}} \left (3038362027 i+2015505180 \sqrt {3}\right )\right ) A+\left (948018096198827 i-433351142966943 \sqrt {3}-\left (30201213892877 i-7860364011 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}-6715196 \left (18980 i-35397 \sqrt {3}\right ) \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) B+\left (5480972498267759 i-1276678374486534 \sqrt {3}-\left (141593468473406 i-24098207903307 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}-28614008 \left (18980 i-35397 \sqrt {3}\right ) \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) C\right )}{\left (18980+35397 i \sqrt {3}\right )^{4/3}}-18 \left (70 \left (367087+7592 \sqrt [3]{18980+35397 i \sqrt {3}}+229 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) A+\left (58757965-2432374 \sqrt [3]{18980+35397 i \sqrt {3}}+36655 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) B+2 \left (125186285-8151059 \sqrt [3]{18980+35397 i \sqrt {3}}+78095 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) C\right ) \left (x-\frac {53}{18}\right )\right )}{11799 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}+\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \int \frac {76527504 \left (\frac {140 \left (2393123977381066087-610744264448441058 i \sqrt {3}+229 \sqrt [3]{18980+35397 i \sqrt {3}} \left (252344979622502+76945470016599 i \sqrt {3}\right )+367087 i \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (3398602427 i+1343670120 \sqrt {3}\right )\right ) A+\left (388238869187458412270-368409390099231280653 i \sqrt {3}+\sqrt [3]{18980+35397 i \sqrt {3}} \left (8563445291472711853+9569151689907273210 i \sqrt {3}\right )[phone] i \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (3398602427 i+1343670120 \sqrt {3}\right )\right ) B+\left (1039212068113121766920-1851256341666229923621 i \sqrt {3}+\sqrt [3]{18980+35397 i \sqrt {3}} \left (20315729875965595021+47169426145839776460 i \sqrt {3}\right )[phone] i \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (3398602427 i+1343670120 \sqrt {3}\right )\right ) C}{\left (18980+35397 i \sqrt {3}\right )^{8/3}}-\frac {18 \left (70 \left (367087+7592 \sqrt [3]{18980+35397 i \sqrt {3}}+229 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) A+\left (58757965-2432374 \sqrt [3]{18980+35397 i \sqrt {3}}+36655 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) B+2 \left (125186285-8151059 \sqrt [3]{18980+35397 i \sqrt {3}}+78095 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) C\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )}{\left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{5/2} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}d\left (x-\frac {53}{18}\right )}{33442519248 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}\right )}{637146 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}-\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (9 (3206 A+7331 B+31238 C)-\frac {\left (\left (2569609+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) (9 B+53 C)-\sqrt [3]{18980+35397 i \sqrt {3}} \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (162 A+477 B+2206 C)\right ) \left (x-\frac {53}{18}\right )}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )}{318573 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{3/2}}\right )}{\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{5/2}}-\frac {C}{27 \left (6 x^3-53 x^2+67 x+70\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {13122 \sqrt {6} \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{5/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{5/2} \left (\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (\frac {i \left (\frac {7 i \left (2 \left (13674725584000 i-171306047863719 \sqrt {3}-1468348 \left (18980 i-35397 \sqrt {3}\right ) \left (18980+35397 i \sqrt {3}\right )^{2/3}-1603 \sqrt [3]{18980+35397 i \sqrt {3}} \left (3038362027 i+2015505180 \sqrt {3}\right )\right ) A+\left (948018096198827 i-433351142966943 \sqrt {3}-\left (30201213892877 i-7860364011 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}-6715196 \left (18980 i-35397 \sqrt {3}\right ) \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) B+\left (5480972498267759 i-1276678374486534 \sqrt {3}-\left (141593468473406 i-24098207903307 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}-28614008 \left (18980 i-35397 \sqrt {3}\right ) \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) C\right )}{\left (18980+35397 i \sqrt {3}\right )^{4/3}}-18 \left (70 \left (367087+7592 \sqrt [3]{18980+35397 i \sqrt {3}}+229 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) A+\left (58757965-2432374 \sqrt [3]{18980+35397 i \sqrt {3}}+36655 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) B+2 \left (125186285-8151059 \sqrt [3]{18980+35397 i \sqrt {3}}+78095 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) C\right ) \left (x-\frac {53}{18}\right )\right )}{11799 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}+\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \int \frac {\frac {140 \left (2393123977381066087-610744264448441058 i \sqrt {3}+229 \sqrt [3]{18980+35397 i \sqrt {3}} \left (252344979622502+76945470016599 i \sqrt {3}\right )+367087 i \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (3398602427 i+1343670120 \sqrt {3}\right )\right ) A+\left (388238869187458412270-368409390099231280653 i \sqrt {3}+\sqrt [3]{18980+35397 i \sqrt {3}} \left (8563445291472711853+9569151689907273210 i \sqrt {3}\right )[phone] i \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (3398602427 i+1343670120 \sqrt {3}\right )\right ) B+\left (1039212068113121766920-1851256341666229923621 i \sqrt {3}+\sqrt [3]{18980+35397 i \sqrt {3}} \left (20315729875965595021+47169426145839776460 i \sqrt {3}\right )[phone] i \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (3398602427 i+1343670120 \sqrt {3}\right )\right ) C}{\left (18980+35397 i \sqrt {3}\right )^{8/3}}-\frac {18 \left (70 \left (367087+7592 \sqrt [3]{18980+35397 i \sqrt {3}}+229 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) A+\left (58757965-2432374 \sqrt [3]{18980+35397 i \sqrt {3}}+36655 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) B+2 \left (125186285-8151059 \sqrt [3]{18980+35397 i \sqrt {3}}+78095 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) C\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}}{\left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{5/2} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}d\left (x-\frac {53}{18}\right )}{437 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}\right )}{637146 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}-\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (9 (3206 A+7331 B+31238 C)-\frac {\left (\left (2569609+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) (9 B+53 C)-\sqrt [3]{18980+35397 i \sqrt {3}} \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (162 A+477 B+2206 C)\right ) \left (x-\frac {53}{18}\right )}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )}{318573 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{3/2}}\right )}{\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{5/2}}-\frac {C}{27 \left (6 x^3-53 x^2+67 x+70\right )^{3/2}}\)

\(\Big \downarrow \) 1237

\(\displaystyle \frac {13122 \sqrt {6} \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{5/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{5/2} \left (\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (\frac {i \left (\frac {7 i \left (2 \left (13674725584000 i-171306047863719 \sqrt {3}-1468348 \left (18980 i-35397 \sqrt {3}\right ) \left (18980+35397 i \sqrt {3}\right )^{2/3}-1603 \sqrt [3]{18980+35397 i \sqrt {3}} \left (3038362027 i+2015505180 \sqrt {3}\right )\right ) A+\left (948018096198827 i-433351142966943 \sqrt {3}-\left (30201213892877 i-7860364011 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}-6715196 \left (18980 i-35397 \sqrt {3}\right ) \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) B+\left (5480972498267759 i-1276678374486534 \sqrt {3}-\left (141593468473406 i-24098207903307 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}-28614008 \left (18980 i-35397 \sqrt {3}\right ) \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) C\right )}{\left (18980+35397 i \sqrt {3}\right )^{4/3}}-18 \left (70 \left (367087+7592 \sqrt [3]{18980+35397 i \sqrt {3}}+229 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) A+\left (58757965-2432374 \sqrt [3]{18980+35397 i \sqrt {3}}+36655 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) B+2 \left (125186285-8151059 \sqrt [3]{18980+35397 i \sqrt {3}}+78095 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) C\right ) \left (x-\frac {53}{18}\right )\right )}{11799 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}+\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (\frac {\left (70 \left (928790925287511109 i+143606221350718932 \sqrt {3}+229 \left (99166270489208 i-19697822934093 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}\right ) A+\left (\left (322951256564282651 i-2031450127312645755 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}+7 \left (3244183065369761005 i+11516686720074975393 \sqrt {3}\right )\right ) B-\left (108269200283003579750 i-437326020453375552807 \sqrt {3}+\sqrt [3]{18980+35397 i \sqrt {3}} \left (4015151831794462393 i+10787669704760845950 \sqrt {3}\right )\right ) C\right ) \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}{9 \left (18980 i-35397 \sqrt {3}\right )^2 \left (2569609 i+\left (18980 i-35397 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}+1603 i \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2}}-\frac {\left (18980+35397 i \sqrt {3}\right )^{2/3} \int \frac {1458 \left (\frac {70 \left (367087 i \left (14944461994741879035714050639 i-13481926216208554875241638492 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}[phone] \left (15968490061977370918974910717+39282393741454535319169624440 i \sqrt {3}\right )-\left (18980+35397 i \sqrt {3}\right )^{2/3} \left (233277777766761512389480320089944+58293093189853523544755229421011 i \sqrt {3}\right )\right ) A+\left (1671781148 i \left (15968490061977370918974910717 i-39282393741454535319169624440 \sqrt {3}\right )+1603 \sqrt [3]{18980+35397 i \sqrt {3}} \left (510816363479200978004550606062155-451513445983553527011978748508169 i \sqrt {3}\right )+7 i \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (3947424346979342928168840582395401 i+2078615189317619467703791945496745 \sqrt {3}\right )\right ) B+\left (13468881836 i \left (15968490061977370918974910717 i-39282393741454535319169624440 \sqrt {3}\right )+1603 \sqrt [3]{18980+35397 i \sqrt {3}} \left (3899835341987112731757200653854490-1854483427710362435650637510281833 i \sqrt {3}\right )+7 i \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (14562826093115563171385849292980557 i+14410532361381640241754886107485130 \sqrt {3}\right )\right ) C}{\left (18980 i-35397 \sqrt {3}\right )^6}-\frac {18 \left (70 i \left (928790925287511109 i+143606221350718932 \sqrt {3}+229 \left (99166270489208 i-19697822934093 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}\right ) A-\left (22709281457588327035-80616807040524827751 i \sqrt {3}+\sqrt [3]{18980+35397 i \sqrt {3}} \left (322951256564282651+2031450127312645755 i \sqrt {3}\right )\right ) B+\left (\sqrt [3]{18980+35397 i \sqrt {3}} \left (4015151831794462393-10787669704760845950 i \sqrt {3}\right )+7 \left (15467028611857654250+62475145779053650401 i \sqrt {3}\right )\right ) C\right ) \left (x-\frac {53}{18}\right )}{\left (18980+35397 i \sqrt {3}\right )^{8/3}}\right )}{\left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}d\left (x-\frac {53}{18}\right )}{1458 \left (2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right )}\right )}{437 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}\right )}{637146 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}-\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (9 (3206 A+7331 B+31238 C)-\frac {\left (\left (2569609+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) (9 B+53 C)-\sqrt [3]{18980+35397 i \sqrt {3}} \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (162 A+477 B+2206 C)\right ) \left (x-\frac {53}{18}\right )}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )}{318573 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{3/2}}\right )}{\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{5/2}}-\frac {C}{27 \left (6 x^3-53 x^2+67 x+70\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {13122 \sqrt {6} \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{5/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{5/2} \left (\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (\frac {i \left (\frac {7 i \left (2 \left (13674725584000 i-171306047863719 \sqrt {3}-1468348 \left (18980 i-35397 \sqrt {3}\right ) \left (18980+35397 i \sqrt {3}\right )^{2/3}-1603 \sqrt [3]{18980+35397 i \sqrt {3}} \left (3038362027 i+2015505180 \sqrt {3}\right )\right ) A+\left (948018096198827 i-433351142966943 \sqrt {3}-\left (30201213892877 i-7860364011 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}-6715196 \left (18980 i-35397 \sqrt {3}\right ) \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) B+\left (5480972498267759 i-1276678374486534 \sqrt {3}-\left (141593468473406 i-24098207903307 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}-28614008 \left (18980 i-35397 \sqrt {3}\right ) \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) C\right )}{\left (18980+35397 i \sqrt {3}\right )^{4/3}}-18 \left (70 \left (367087+7592 \sqrt [3]{18980+35397 i \sqrt {3}}+229 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) A+\left (58757965-2432374 \sqrt [3]{18980+35397 i \sqrt {3}}+36655 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) B+2 \left (125186285-8151059 \sqrt [3]{18980+35397 i \sqrt {3}}+78095 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) C\right ) \left (x-\frac {53}{18}\right )\right )}{11799 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}+\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (\frac {\left (70 \left (928790925287511109 i+143606221350718932 \sqrt {3}+229 \left (99166270489208 i-19697822934093 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}\right ) A+\left (\left (322951256564282651 i-2031450127312645755 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}+7 \left (3244183065369761005 i+11516686720074975393 \sqrt {3}\right )\right ) B-\left (108269200283003579750 i-437326020453375552807 \sqrt {3}+\sqrt [3]{18980+35397 i \sqrt {3}} \left (4015151831794462393 i+10787669704760845950 \sqrt {3}\right )\right ) C\right ) \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}{9 \left (18980 i-35397 \sqrt {3}\right )^2 \left (2569609 i+\left (18980 i-35397 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}+1603 i \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2}}-\frac {\left (18980+35397 i \sqrt {3}\right )^{2/3} \int \frac {\frac {70 \left (367087 i \left (14944461994741879035714050639 i-13481926216208554875241638492 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}[phone] \left (15968490061977370918974910717+39282393741454535319169624440 i \sqrt {3}\right )-\left (18980+35397 i \sqrt {3}\right )^{2/3} \left (233277777766761512389480320089944+58293093189853523544755229421011 i \sqrt {3}\right )\right ) A+\left (1671781148 i \left (15968490061977370918974910717 i-39282393741454535319169624440 \sqrt {3}\right )+1603 \sqrt [3]{18980+35397 i \sqrt {3}} \left (510816363479200978004550606062155-451513445983553527011978748508169 i \sqrt {3}\right )+7 i \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (3947424346979342928168840582395401 i+2078615189317619467703791945496745 \sqrt {3}\right )\right ) B+\left (13468881836 i \left (15968490061977370918974910717 i-39282393741454535319169624440 \sqrt {3}\right )+1603 \sqrt [3]{18980+35397 i \sqrt {3}} \left (3899835341987112731757200653854490-1854483427710362435650637510281833 i \sqrt {3}\right )+7 i \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (14562826093115563171385849292980557 i+14410532361381640241754886107485130 \sqrt {3}\right )\right ) C}{\left (18980 i-35397 \sqrt {3}\right )^6}-\frac {18 \left (70 i \left (928790925287511109 i+143606221350718932 \sqrt {3}+229 \left (99166270489208 i-19697822934093 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}\right ) A-\left (22709281457588327035-80616807040524827751 i \sqrt {3}+\sqrt [3]{18980+35397 i \sqrt {3}} \left (322951256564282651+2031450127312645755 i \sqrt {3}\right )\right ) B+\left (\sqrt [3]{18980+35397 i \sqrt {3}} \left (4015151831794462393-10787669704760845950 i \sqrt {3}\right )+7 \left (15467028611857654250+62475145779053650401 i \sqrt {3}\right )\right ) C\right ) \left (x-\frac {53}{18}\right )}{\left (18980+35397 i \sqrt {3}\right )^{8/3}}}{\left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}d\left (x-\frac {53}{18}\right )}{2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}}\right )}{437 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}\right )}{637146 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}-\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (9 (3206 A+7331 B+31238 C)-\frac {\left (\left (2569609+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) (9 B+53 C)-\sqrt [3]{18980+35397 i \sqrt {3}} \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (162 A+477 B+2206 C)\right ) \left (x-\frac {53}{18}\right )}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )}{318573 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{3/2}}\right )}{\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{5/2}}-\frac {C}{27 \left (6 x^3-53 x^2+67 x+70\right )^{3/2}}\)

\(\Big \downarrow \) 1237

\(\displaystyle \frac {13122 \sqrt {6} \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{5/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{5/2} \left (\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (\frac {i \left (\frac {7 i \left (2 \left (13674725584000 i-171306047863719 \sqrt {3}-1468348 \left (18980 i-35397 \sqrt {3}\right ) \left (18980+35397 i \sqrt {3}\right )^{2/3}-1603 \sqrt [3]{18980+35397 i \sqrt {3}} \left (3038362027 i+2015505180 \sqrt {3}\right )\right ) A+\left (948018096198827 i-433351142966943 \sqrt {3}-\left (30201213892877 i-7860364011 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}-6715196 \left (18980 i-35397 \sqrt {3}\right ) \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) B+\left (5480972498267759 i-1276678374486534 \sqrt {3}-\left (141593468473406 i-24098207903307 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}-28614008 \left (18980 i-35397 \sqrt {3}\right ) \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) C\right )}{\left (18980+35397 i \sqrt {3}\right )^{4/3}}-18 \left (70 \left (367087+7592 \sqrt [3]{18980+35397 i \sqrt {3}}+229 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) A+\left (58757965-2432374 \sqrt [3]{18980+35397 i \sqrt {3}}+36655 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) B+2 \left (125186285-8151059 \sqrt [3]{18980+35397 i \sqrt {3}}+78095 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) C\right ) \left (x-\frac {53}{18}\right )\right )}{11799 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}+\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (\frac {\left (70 \left (928790925287511109 i+143606221350718932 \sqrt {3}+229 \left (99166270489208 i-19697822934093 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}\right ) A+\left (\left (322951256564282651 i-2031450127312645755 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}+7 \left (3244183065369761005 i+11516686720074975393 \sqrt {3}\right )\right ) B-\left (108269200283003579750 i-437326020453375552807 \sqrt {3}+\sqrt [3]{18980+35397 i \sqrt {3}} \left (4015151831794462393 i+10787669704760845950 \sqrt {3}\right )\right ) C\right ) \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}{9 \left (18980 i-35397 \sqrt {3}\right )^2 \left (2569609 i+\left (18980 i-35397 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}+1603 i \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2}}-\frac {\left (18980+35397 i \sqrt {3}\right )^{2/3} \left (\frac {2 \left (18980+35397 i \sqrt {3}\right )^{2/3} (243399520 A-417945287 B-3367220459 C) \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}{9 \left (2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}}}-\frac {\left (18980+35397 i \sqrt {3}\right )^{2/3} \int \frac {972 \left (\frac {35 \left (7863084987548729297234369754226826745999620 i+9652736723986449646832212205360079442026123 \sqrt {3}\right ) (440325694 A+677197987 B+2348953618 C)}{\left (18980 i-35397 \sqrt {3}\right )^9}[phone] A-417945287 B-3367220459 C) \left (x-\frac {53}{18}\right )\right )}{\sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}d\left (x-\frac {53}{18}\right )}{486 \left (2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right )}\right )}{2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}}\right )}{437 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}\right )}{637146 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}-\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (9 (3206 A+7331 B+31238 C)-\frac {\left (\left (2569609+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) (9 B+53 C)-\sqrt [3]{18980+35397 i \sqrt {3}} \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (162 A+477 B+2206 C)\right ) \left (x-\frac {53}{18}\right )}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )}{318573 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{3/2}}\right )}{\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{5/2}}-\frac {C}{27 \left (6 x^3-53 x^2+67 x+70\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {13122 \sqrt {6} \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{5/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{5/2} \left (\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (\frac {i \left (\frac {7 i \left (2 \left (13674725584000 i-171306047863719 \sqrt {3}-1468348 \left (18980 i-35397 \sqrt {3}\right ) \left (18980+35397 i \sqrt {3}\right )^{2/3}-1603 \sqrt [3]{18980+35397 i \sqrt {3}} \left (3038362027 i+2015505180 \sqrt {3}\right )\right ) A+\left (948018096198827 i-433351142966943 \sqrt {3}-\left (30201213892877 i-7860364011 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}-6715196 \left (18980 i-35397 \sqrt {3}\right ) \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) B+\left (5480972498267759 i-1276678374486534 \sqrt {3}-\left (141593468473406 i-24098207903307 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}-28614008 \left (18980 i-35397 \sqrt {3}\right ) \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) C\right )}{\left (18980+35397 i \sqrt {3}\right )^{4/3}}-18 \left (70 \left (367087+7592 \sqrt [3]{18980+35397 i \sqrt {3}}+229 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) A+\left (58757965-2432374 \sqrt [3]{18980+35397 i \sqrt {3}}+36655 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) B+2 \left (125186285-8151059 \sqrt [3]{18980+35397 i \sqrt {3}}+78095 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) C\right ) \left (x-\frac {53}{18}\right )\right )}{11799 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}+\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (\frac {\left (70 \left (928790925287511109 i+143606221350718932 \sqrt {3}+229 \left (99166270489208 i-19697822934093 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}\right ) A+\left (\left (322951256564282651 i-2031450127312645755 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}+7 \left (3244183065369761005 i+11516686720074975393 \sqrt {3}\right )\right ) B-\left (108269200283003579750 i-437326020453375552807 \sqrt {3}+\sqrt [3]{18980+35397 i \sqrt {3}} \left (4015151831794462393 i+10787669704760845950 \sqrt {3}\right )\right ) C\right ) \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}{9 \left (18980 i-35397 \sqrt {3}\right )^2 \left (2569609 i+\left (18980 i-35397 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}+1603 i \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2}}-\frac {\left (18980+35397 i \sqrt {3}\right )^{2/3} \left (\frac {2 \left (18980+35397 i \sqrt {3}\right )^{2/3} (243399520 A-417945287 B-3367220459 C) \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}{9 \left (2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}}}-\frac {2 \left (18980+35397 i \sqrt {3}\right )^{2/3} \int \frac {\frac {35 \left (7863084987548729297234369754226826745999620 i+9652736723986449646832212205360079442026123 \sqrt {3}\right ) (440325694 A+677197987 B+2348953618 C)}{\left (18980 i-35397 \sqrt {3}\right )^9}[phone] A-417945287 B-3367220459 C) \left (x-\frac {53}{18}\right )}{\sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}d\left (x-\frac {53}{18}\right )}{2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}}\right )}{2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}}\right )}{437 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}\right )}{637146 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}-\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (9 (3206 A+7331 B+31238 C)-\frac {\left (\left (2569609+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) (9 B+53 C)-\sqrt [3]{18980+35397 i \sqrt {3}} \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (162 A+477 B+2206 C)\right ) \left (x-\frac {53}{18}\right )}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )}{318573 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{3/2}}\right )}{\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{5/2}}-\frac {C}{27 \left (6 x^3-53 x^2+67 x+70\right )^{3/2}}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {13122 \sqrt {6} \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{5/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{5/2} \left (\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (\frac {i \left (\frac {7 i \left (2 \left (13674725584000 i-171306047863719 \sqrt {3}-1468348 \left (18980 i-35397 \sqrt {3}\right ) \left (18980+35397 i \sqrt {3}\right )^{2/3}-1603 \sqrt [3]{18980+35397 i \sqrt {3}} \left (3038362027 i+2015505180 \sqrt {3}\right )\right ) A+\left (948018096198827 i-433351142966943 \sqrt {3}-\left (30201213892877 i-7860364011 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}-6715196 \left (18980 i-35397 \sqrt {3}\right ) \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) B+\left (5480972498267759 i-1276678374486534 \sqrt {3}-\left (141593468473406 i-24098207903307 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}-28614008 \left (18980 i-35397 \sqrt {3}\right ) \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) C\right )}{\left (18980+35397 i \sqrt {3}\right )^{4/3}}-18 \left (70 \left (367087+7592 \sqrt [3]{18980+35397 i \sqrt {3}}+229 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) A+\left (58757965-2432374 \sqrt [3]{18980+35397 i \sqrt {3}}+36655 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) B+2 \left (125186285-8151059 \sqrt [3]{18980+35397 i \sqrt {3}}+78095 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) C\right ) \left (x-\frac {53}{18}\right )\right )}{11799 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}+\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (\frac {\left (70 \left (928790925287511109 i+143606221350718932 \sqrt {3}+229 \left (99166270489208 i-19697822934093 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}\right ) A+\left (\left (322951256564282651 i-2031450127312645755 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}+7 \left (3244183065369761005 i+11516686720074975393 \sqrt {3}\right )\right ) B-\left (108269200283003579750 i-437326020453375552807 \sqrt {3}+\sqrt [3]{18980+35397 i \sqrt {3}} \left (4015151831794462393 i+10787669704760845950 \sqrt {3}\right )\right ) C\right ) \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}{9 \left (18980 i-35397 \sqrt {3}\right )^2 \left (2569609 i+\left (18980 i-35397 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}+1603 i \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2}}-\frac {\left (18980+35397 i \sqrt {3}\right )^{2/3} \left (\frac {2 \left (18980+35397 i \sqrt {3}\right )^{2/3} (243399520 A-417945287 B-3367220459 C) \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}{9 \left (2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}}}-\frac {2 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (\left (\frac {\left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (243399520 A-417945287 B-3367220459 C)}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\frac {35 \left (7863084987548729297234369754226826745999620 i+9652736723986449646832212205360079442026123 \sqrt {3}\right ) (440325694 A+677197987 B+2348953618 C)}{\left (18980 i-35397 \sqrt {3}\right )^9}\right ) \int \frac {1}{\sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}d\left (x-\frac {53}{18}\right )+(243399520 A-417945287 B-3367220459 C) \int \frac {\sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}}}{\sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}d\left (x-\frac {53}{18}\right )\right )}{2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}}\right )}{2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}}\right )}{437 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}\right )}{637146 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}-\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (9 (3206 A+7331 B+31238 C)-\frac {\left (\left (2569609+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) (9 B+53 C)-\sqrt [3]{18980+35397 i \sqrt {3}} \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (162 A+477 B+2206 C)\right ) \left (x-\frac {53}{18}\right )}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )}{318573 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{3/2}}\right )}{\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{5/2}}-\frac {C}{27 \left (6 x^3-53 x^2+67 x+70\right )^{3/2}}\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {13122 \sqrt {6} \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{5/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{5/2} \left (\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (\frac {i \left (\frac {7 i \left (2 \left (13674725584000 i-171306047863719 \sqrt {3}-1468348 \left (18980 i-35397 \sqrt {3}\right ) \left (18980+35397 i \sqrt {3}\right )^{2/3}-1603 \sqrt [3]{18980+35397 i \sqrt {3}} \left (3038362027 i+2015505180 \sqrt {3}\right )\right ) A+\left (948018096198827 i-433351142966943 \sqrt {3}-\left (30201213892877 i-7860364011 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}-6715196 \left (18980 i-35397 \sqrt {3}\right ) \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) B+\left (5480972498267759 i-1276678374486534 \sqrt {3}-\left (141593468473406 i-24098207903307 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}-28614008 \left (18980 i-35397 \sqrt {3}\right ) \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) C\right )}{\left (18980+35397 i \sqrt {3}\right )^{4/3}}-18 \left (70 \left (367087+7592 \sqrt [3]{18980+35397 i \sqrt {3}}+229 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) A+\left (58757965-2432374 \sqrt [3]{18980+35397 i \sqrt {3}}+36655 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) B+2 \left (125186285-8151059 \sqrt [3]{18980+35397 i \sqrt {3}}+78095 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) C\right ) \left (x-\frac {53}{18}\right )\right )}{11799 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}+\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (\frac {\left (70 \left (928790925287511109 i+143606221350718932 \sqrt {3}+229 \left (99166270489208 i-19697822934093 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}\right ) A+\left (\left (322951256564282651 i-2031450127312645755 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}+7 \left (3244183065369761005 i+11516686720074975393 \sqrt {3}\right )\right ) B-\left (108269200283003579750 i-437326020453375552807 \sqrt {3}+\sqrt [3]{18980+35397 i \sqrt {3}} \left (4015151831794462393 i+10787669704760845950 \sqrt {3}\right )\right ) C\right ) \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}{9 \left (18980 i-35397 \sqrt {3}\right )^2 \left (2569609 i+\left (18980 i-35397 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}+1603 i \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2}}-\frac {\left (18980+35397 i \sqrt {3}\right )^{2/3} \left (\frac {2 \left (18980+35397 i \sqrt {3}\right )^{2/3} (243399520 A-417945287 B-3367220459 C) \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}{9 \left (2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}}}-\frac {2 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (-\frac {\sqrt {\frac {2}{3}} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (\frac {\left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (243399520 A-417945287 B-3367220459 C)}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\frac {35 \left (7863084987548729297234369754226826745999620 i+9652736723986449646832212205360079442026123 \sqrt {3}\right ) (440325694 A+677197987 B+2348953618 C)}{\left (18980 i-35397 \sqrt {3}\right )^9}\right ) \sqrt {-\frac {\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}-18 \left (x-\frac {53}{18}\right )}{\frac {1}{324} \sqrt {-\frac {1}{3} \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (324-\frac {519372}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}}} \sqrt {-\frac {\left (18980+35397 i \sqrt {3}\right )^{2/3} \left (-324 \left (x-\frac {53}{18}\right )^2-\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}-\left (18980+35397 i \sqrt {3}\right )^{2/3}-\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}+1603\right )}{\left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )^2}} \int \frac {1}{\sqrt {1-\frac {\sqrt {-\frac {1}{3} \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (-36 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\sqrt {-3 \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (1-\frac {1603}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )\right )}{2 \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}} \sqrt {\frac {\sqrt {-\left (18980+35397 i \sqrt {3}\right )^{2/3}} \sqrt [3]{18980+35397 i \sqrt {3}} \left (-36 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\sqrt {-3 \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (1-\frac {1603}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )\right )}{\sqrt {3} \left (18980+35397 i \sqrt {3}-1603 \sqrt [3]{18980+35397 i \sqrt {3}}+\sqrt {-3 \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )\right )}+1}}d\frac {\sqrt {\frac {\sqrt {-3 \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (-36 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\sqrt {-3 \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (1-\frac {1603}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )\right )}{1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}}}}{\sqrt {6}}}{9 \sqrt {-\left (18980+35397 i \sqrt {3}\right )^{2/3}} \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}-\frac {\left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (243399520 A-417945287 B-3367220459 C) \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \sqrt {-\frac {\left (18980+35397 i \sqrt {3}\right )^{2/3} \left (-324 \left (x-\frac {53}{18}\right )^2-\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}-\left (18980+35397 i \sqrt {3}\right )^{2/3}-\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}+1603\right )}{\left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )^2}} \int \frac {\sqrt {\frac {\sqrt {-\left (18980+35397 i \sqrt {3}\right )^{2/3}} \sqrt [3]{18980+35397 i \sqrt {3}} \left (-36 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\sqrt {-3 \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (1-\frac {1603}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )\right )}{\sqrt {3} \left (18980+35397 i \sqrt {3}-1603 \sqrt [3]{18980+35397 i \sqrt {3}}+\sqrt {-3 \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )\right )}+1}}{\sqrt {1-\frac {\sqrt {-\frac {1}{3} \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (-36 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\sqrt {-3 \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (1-\frac {1603}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )\right )}{2 \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}}}d\frac {\sqrt {\frac {\sqrt {-3 \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (-36 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\sqrt {-3 \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (1-\frac {1603}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )\right )}{1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}}}}{\sqrt {6}}}{3 \sqrt {6} \sqrt {-\left (18980+35397 i \sqrt {3}\right )^{2/3}} \sqrt {-\frac {\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}-18 \left (x-\frac {53}{18}\right )}{\frac {1}{324} \sqrt {-\frac {1}{3} \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (324-\frac {519372}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}\right )}{2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}}\right )}{2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}}\right )}{437 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}\right )}{637146 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}-\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (9 (3206 A+7331 B+31238 C)-\frac {\left (\left (2569609+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) (9 B+53 C)-\sqrt [3]{18980+35397 i \sqrt {3}} \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (162 A+477 B+2206 C)\right ) \left (x-\frac {53}{18}\right )}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )}{318573 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{3/2}}\right )}{\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{5/2}}-\frac {C}{27 \left (6 x^3-53 x^2+67 x+70\right )^{3/2}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {13122 \sqrt {6} \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{5/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{5/2} \left (\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (\frac {i \left (\frac {7 i \left (2 \left (13674725584000 i-171306047863719 \sqrt {3}-1468348 \left (18980 i-35397 \sqrt {3}\right ) \left (18980+35397 i \sqrt {3}\right )^{2/3}-1603 \sqrt [3]{18980+35397 i \sqrt {3}} \left (3038362027 i+2015505180 \sqrt {3}\right )\right ) A+\left (948018096198827 i-433351142966943 \sqrt {3}-\left (30201213892877 i-7860364011 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}-6715196 \left (18980 i-35397 \sqrt {3}\right ) \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) B+\left (5480972498267759 i-1276678374486534 \sqrt {3}-\left (141593468473406 i-24098207903307 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}-28614008 \left (18980 i-35397 \sqrt {3}\right ) \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) C\right )}{\left (18980+35397 i \sqrt {3}\right )^{4/3}}-18 \left (70 \left (367087+7592 \sqrt [3]{18980+35397 i \sqrt {3}}+229 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) A+\left (58757965-2432374 \sqrt [3]{18980+35397 i \sqrt {3}}+36655 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) B+2 \left (125186285-8151059 \sqrt [3]{18980+35397 i \sqrt {3}}+78095 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) C\right ) \left (x-\frac {53}{18}\right )\right )}{11799 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}+\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (\frac {\left (70 \left (928790925287511109 i+143606221350718932 \sqrt {3}+229 \left (99166270489208 i-19697822934093 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}\right ) A+\left (\left (322951256564282651 i-2031450127312645755 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}+7 \left (3244183065369761005 i+11516686720074975393 \sqrt {3}\right )\right ) B-\left (108269200283003579750 i-437326020453375552807 \sqrt {3}+\sqrt [3]{18980+35397 i \sqrt {3}} \left (4015151831794462393 i+10787669704760845950 \sqrt {3}\right )\right ) C\right ) \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}{9 \left (18980 i-35397 \sqrt {3}\right )^2 \left (2569609 i+\left (18980 i-35397 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}+1603 i \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2}}-\frac {\left (18980+35397 i \sqrt {3}\right )^{2/3} \left (\frac {2 \left (18980+35397 i \sqrt {3}\right )^{2/3} (243399520 A-417945287 B-3367220459 C) \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}{9 \left (2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}}}-\frac {2 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (\frac {\sqrt {\frac {2}{3}} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (\frac {\left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (243399520 A-417945287 B-3367220459 C)}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\frac {35 \left (7863084987548729297234369754226826745999620 i+9652736723986449646832212205360079442026123 \sqrt {3}\right ) (440325694 A+677197987 B+2348953618 C)}{\left (18980 i-35397 \sqrt {3}\right )^9}\right ) \sqrt {-\frac {\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}-18 \left (x-\frac {53}{18}\right )}{\frac {1}{324} \sqrt {-\frac {1}{3} \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (324-\frac {519372}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}}} \sqrt {-\frac {\left (18980+35397 i \sqrt {3}\right )^{2/3} \left (-324 \left (x-\frac {53}{18}\right )^2-\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}-\left (18980+35397 i \sqrt {3}\right )^{2/3}-\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}+1603\right )}{\left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {53}{18}-x\right ),\frac {2 \left (18980+35397 i \sqrt {3}-1603 \sqrt [3]{18980+35397 i \sqrt {3}}\right )}{18980+35397 i \sqrt {3}-1603 \sqrt [3]{18980+35397 i \sqrt {3}}+\sqrt {-3 \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}\right )}{9 \sqrt {-\left (18980+35397 i \sqrt {3}\right )^{2/3}} \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}-\frac {\left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (243399520 A-417945287 B-3367220459 C) \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \sqrt {-\frac {\left (18980+35397 i \sqrt {3}\right )^{2/3} \left (-324 \left (x-\frac {53}{18}\right )^2-\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}-\left (18980+35397 i \sqrt {3}\right )^{2/3}-\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}+1603\right )}{\left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )^2}} \int \frac {\sqrt {\frac {\sqrt {-\left (18980+35397 i \sqrt {3}\right )^{2/3}} \sqrt [3]{18980+35397 i \sqrt {3}} \left (-36 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\sqrt {-3 \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (1-\frac {1603}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )\right )}{\sqrt {3} \left (18980+35397 i \sqrt {3}-1603 \sqrt [3]{18980+35397 i \sqrt {3}}+\sqrt {-3 \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )\right )}+1}}{\sqrt {1-\frac {\sqrt {-\frac {1}{3} \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (-36 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\sqrt {-3 \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (1-\frac {1603}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )\right )}{2 \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}}}d\frac {\sqrt {\frac {\sqrt {-3 \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (-36 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\sqrt {-3 \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (1-\frac {1603}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )\right )}{1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}}}}{\sqrt {6}}}{3 \sqrt {6} \sqrt {-\left (18980+35397 i \sqrt {3}\right )^{2/3}} \sqrt {-\frac {\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}-18 \left (x-\frac {53}{18}\right )}{\frac {1}{324} \sqrt {-\frac {1}{3} \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (324-\frac {519372}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}\right )}{2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}}\right )}{2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}}\right )}{437 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}\right )}{637146 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}-\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (9 (3206 A+7331 B+31238 C)-\frac {\left (\left (2569609+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) (9 B+53 C)-\sqrt [3]{18980+35397 i \sqrt {3}} \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (162 A+477 B+2206 C)\right ) \left (x-\frac {53}{18}\right )}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )}{318573 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{3/2}}\right )}{\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{5/2}}-\frac {C}{27 \left (6 x^3-53 x^2+67 x+70\right )^{3/2}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {13122 \sqrt {6} \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{5/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{5/2} \left (\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (\frac {i \left (\frac {7 i \left (2 \left (13674725584000 i-171306047863719 \sqrt {3}-1468348 \left (18980 i-35397 \sqrt {3}\right ) \left (18980+35397 i \sqrt {3}\right )^{2/3}-1603 \sqrt [3]{18980+35397 i \sqrt {3}} \left (3038362027 i+2015505180 \sqrt {3}\right )\right ) A+\left (948018096198827 i-433351142966943 \sqrt {3}-\left (30201213892877 i-7860364011 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}-6715196 \left (18980 i-35397 \sqrt {3}\right ) \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) B+\left (5480972498267759 i-1276678374486534 \sqrt {3}-\left (141593468473406 i-24098207903307 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}-28614008 \left (18980 i-35397 \sqrt {3}\right ) \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) C\right )}{\left (18980+35397 i \sqrt {3}\right )^{4/3}}-18 \left (70 \left (367087+7592 \sqrt [3]{18980+35397 i \sqrt {3}}+229 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) A+\left (58757965-2432374 \sqrt [3]{18980+35397 i \sqrt {3}}+36655 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) B+2 \left (125186285-8151059 \sqrt [3]{18980+35397 i \sqrt {3}}+78095 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) C\right ) \left (x-\frac {53}{18}\right )\right )}{11799 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}+\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (\frac {\left (70 \left (928790925287511109 i+143606221350718932 \sqrt {3}+229 \left (99166270489208 i-19697822934093 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}\right ) A+\left (\left (322951256564282651 i-2031450127312645755 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}+7 \left (3244183065369761005 i+11516686720074975393 \sqrt {3}\right )\right ) B-\left (108269200283003579750 i-437326020453375552807 \sqrt {3}+\sqrt [3]{18980+35397 i \sqrt {3}} \left (4015151831794462393 i+10787669704760845950 \sqrt {3}\right )\right ) C\right ) \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}{9 \left (18980 i-35397 \sqrt {3}\right )^2 \left (2569609 i+\left (18980 i-35397 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}+1603 i \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2}}-\frac {\left (18980+35397 i \sqrt {3}\right )^{2/3} \left (\frac {2 \left (18980+35397 i \sqrt {3}\right )^{2/3} (243399520 A-417945287 B-3367220459 C) \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}{9 \left (2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}}}-\frac {2 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (\frac {\left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (243399520 A-417945287 B-3367220459 C) \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \sqrt {-\frac {\left (18980+35397 i \sqrt {3}\right )^{2/3} \left (-324 \left (x-\frac {53}{18}\right )^2-\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}-\left (18980+35397 i \sqrt {3}\right )^{2/3}-\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}+1603\right )}{\left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )^2}} E\left (\arcsin \left (\frac {53}{18}-x\right )|\frac {2 \left (18980+35397 i \sqrt {3}-1603 \sqrt [3]{18980+35397 i \sqrt {3}}\right )}{18980+35397 i \sqrt {3}-1603 \sqrt [3]{18980+35397 i \sqrt {3}}+\sqrt {-3 \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}\right )}{3 \sqrt {6} \sqrt {-\left (18980+35397 i \sqrt {3}\right )^{2/3}} \sqrt {-\frac {\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}-18 \left (x-\frac {53}{18}\right )}{\frac {1}{324} \sqrt {-\frac {1}{3} \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (324-\frac {519372}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}+\frac {\sqrt {\frac {2}{3}} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (\frac {\left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (243399520 A-417945287 B-3367220459 C)}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\frac {35 \left (7863084987548729297234369754226826745999620 i+9652736723986449646832212205360079442026123 \sqrt {3}\right ) (440325694 A+677197987 B+2348953618 C)}{\left (18980 i-35397 \sqrt {3}\right )^9}\right ) \sqrt {-\frac {\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}-18 \left (x-\frac {53}{18}\right )}{\frac {1}{324} \sqrt {-\frac {1}{3} \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (324-\frac {519372}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}}} \sqrt {-\frac {\left (18980+35397 i \sqrt {3}\right )^{2/3} \left (-324 \left (x-\frac {53}{18}\right )^2-\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}-\left (18980+35397 i \sqrt {3}\right )^{2/3}-\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}+1603\right )}{\left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {53}{18}-x\right ),\frac {2 \left (18980+35397 i \sqrt {3}-1603 \sqrt [3]{18980+35397 i \sqrt {3}}\right )}{18980+35397 i \sqrt {3}-1603 \sqrt [3]{18980+35397 i \sqrt {3}}+\sqrt {-3 \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}\right )}{9 \sqrt {-\left (18980+35397 i \sqrt {3}\right )^{2/3}} \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}\right )}{2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}}\right )}{2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}}\right )}{437 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}\right )}{637146 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}-\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (9 (3206 A+7331 B+31238 C)-\frac {\left (\left (2569609+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) (9 B+53 C)-\sqrt [3]{18980+35397 i \sqrt {3}} \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (162 A+477 B+2206 C)\right ) \left (x-\frac {53}{18}\right )}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )}{318573 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{3/2}}\right )}{\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{5/2}}-\frac {C}{27 \left (6 x^3-53 x^2+67 x+70\right )^{3/2}}\)

Input:

Int[(A + B*x + C*x^2)/(70 + 67*x - 53*x^2 + 6*x^3)^(5/2),x]
 

Output:

-1/27*C/(70 + 67*x - 53*x^2 + 6*x^3)^(3/2) + (13122*Sqrt[6]*(-((1603 + (18 
980 + (35397*I)*Sqrt[3])^(2/3))/(18980 + (35397*I)*Sqrt[3])^(1/3)) + 18*(- 
53/18 + x))^(5/2)*(-1603 + 2569609/(18980 + (35397*I)*Sqrt[3])^(2/3) + (18 
980 + (35397*I)*Sqrt[3])^(2/3) + (18*(1603 + (18980 + (35397*I)*Sqrt[3])^( 
2/3))*(-53/18 + x))/(18980 + (35397*I)*Sqrt[3])^(1/3) + 324*(-53/18 + x)^2 
)^(5/2)*(((-1/318573*I)*(18980 + (35397*I)*Sqrt[3])^(1/3)*(9*(3206*A + 733 
1*B + 31238*C) - (((2569609 + (18980 + (35397*I)*Sqrt[3])^(4/3))*(9*B + 53 
*C) - (18980 + (35397*I)*Sqrt[3])^(1/3)*(1603 + (18980 + (35397*I)*Sqrt[3] 
)^(2/3))*(162*A + 477*B + 2206*C))*(-53/18 + x))/(18980 + (35397*I)*Sqrt[3 
])^(2/3)))/(Sqrt[3]*(1603 - (18980 + (35397*I)*Sqrt[3])^(2/3))*(-((1603 + 
(18980 + (35397*I)*Sqrt[3])^(2/3))/(18980 + (35397*I)*Sqrt[3])^(1/3)) + 18 
*(-53/18 + x))^(3/2)*(-1603 + 2569609/(18980 + (35397*I)*Sqrt[3])^(2/3) + 
(18980 + (35397*I)*Sqrt[3])^(2/3) + (18*(1603 + (18980 + (35397*I)*Sqrt[3] 
)^(2/3))*(-53/18 + x))/(18980 + (35397*I)*Sqrt[3])^(1/3) + 324*(-53/18 + x 
)^2)^(3/2)) + ((I/637146)*(18980 + (35397*I)*Sqrt[3])^(1/3)*(((I/11799)*(( 
(7*I)*(2*(13674725584000*I - 171306047863719*Sqrt[3] - 1468348*(18980*I - 
35397*Sqrt[3])*(18980 + (35397*I)*Sqrt[3])^(2/3) - 1603*(18980 + (35397*I) 
*Sqrt[3])^(1/3)*(3038362027*I + 2015505180*Sqrt[3]))*A + (948018096198827* 
I - 433351142966943*Sqrt[3] - (30201213892877*I - 7860364011*Sqrt[3])*(189 
80 + (35397*I)*Sqrt[3])^(1/3) - 6715196*(18980*I - 35397*Sqrt[3])*(1898...
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 1172

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sy 
mbol] :> Simp[2*Rt[b^2 - 4*a*c, 2]*(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2 
)/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*e - e 
*Rt[b^2 - 4*a*c, 2])))^m))   Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2* 
c*d - b*e - e*Rt[b^2 - 4*a*c, 2])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^ 
2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b, c, d, e 
}, x] && EqQ[m^2, 1/4]
 

rule 1235

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 
*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a 
+ b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] 
 + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^m 
*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 
 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* 
m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - 
f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, 
 m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] 
)
 

rule 1237

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b* 
x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1) 
*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[ 
(c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m 
+ 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 
] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

rule 2486

Int[((e_.) + (f_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_S 
ymbol] :> With[{r = Rt[-9*a*d^2 + Sqrt[3]*d*Sqrt[4*b^3*d + 27*a^2*d^2], 3]} 
, Simp[(a + b*x + d*x^3)^p/(Simp[18^(1/3)*b*(d/(3*r)) - r/18^(1/3) + d*x, x 
]^p*Simp[b*(d/3) + 12^(1/3)*b^2*(d^2/(3*r^2)) + r^2/(3*12^(1/3)) - d*(2^(1/ 
3)*b*(d/(3^(1/3)*r)) - r/18^(1/3))*x + d^2*x^2, x]^p)   Int[(e + f*x)^m*Sim 
p[18^(1/3)*b*(d/(3*r)) - r/18^(1/3) + d*x, x]^p*Simp[b*(d/3) + 12^(1/3)*b^2 
*(d^2/(3*r^2)) + r^2/(3*12^(1/3)) - d*(2^(1/3)*b*(d/(3^(1/3)*r)) - r/18^(1/ 
3))*x + d^2*x^2, x]^p, x], x]] /; FreeQ[{a, b, d, e, f, m, p}, x] && NeQ[4* 
b^3 + 27*a^2*d, 0] &&  !IntegerQ[p]
 

rule 2490

Int[(P3_)^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> With[{a = Coeff[P3 
, x, 0], b = Coeff[P3, x, 1], c = Coeff[P3, x, 2], d = Coeff[P3, x, 3]}, Su 
bst[Int[((3*d*e - c*f)/(3*d) + f*x)^m*Simp[(2*c^3 - 9*b*c*d + 27*a*d^2)/(27 
*d^2) - (c^2 - 3*b*d)*(x/(3*d)) + d*x^3, x]^p, x], x, x + c/(3*d)] /; NeQ[c 
, 0]] /; FreeQ[{e, f, m, p}, x] && PolyQ[P3, x, 3]
 

rule 2526

Int[(Pm_)*(Qn_)^(p_), x_Symbol] :> With[{m = Expon[Pm, x], n = Expon[Qn, x] 
}, Simp[Coeff[Pm, x, m]*(Qn^(p + 1)/(n*(p + 1)*Coeff[Qn, x, n])), x] + Simp 
[1/(n*Coeff[Qn, x, n])   Int[ExpandToSum[n*Coeff[Qn, x, n]*Pm - Coeff[Pm, x 
, m]*D[Qn, x], x]*Qn^p, x], x] /; EqQ[m, n - 1]] /; FreeQ[p, x] && PolyQ[Pm 
, x] && PolyQ[Qn, x] && NeQ[p, -1]
 

Maple [A] (verified)

Time = 1.13 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.58

method	result	size
elliptic	\(\frac {\left (\left (-\frac {7331 B}{139216401}-\frac {31238 C}{139216401}-\frac {3206 A}{139216401}\right ) x^{2}+\left (\frac {10585 B}{43963074}+\frac {37663 C}{43963074}+\frac {3314 A}{21981537}\right ) x +\frac {112210 B}{417649203}+\frac {256585 C}{417649203}-\frac {13687 A}{835298406}\right ) \sqrt {6 x^{3}-53 x^{2}+67 x +70}}{\left (x^{3}-\frac {53}{6} x^{2}+\frac {67}{6} x +\frac {35}{3}\right )^{2}}-\frac {12 \left (\left (\frac {486799040 A}{717822455829363}-\frac {835890574 B}{717822455829363}-\frac {6734440918 C}{717822455829363}\right ) x^{2}+\left (-\frac {723013130 A}{113340387762531}+\frac {361408261 B}{113340387762531}+\frac {4819508632 C}{113340387762531}\right ) x +\frac {13336565795 A}{2153467367488089}+\frac {26335676756 B}{2153467367488089}+\frac {210892425319 C}{4306934734976178}\right )}{\sqrt {6 x^{3}-53 x^{2}+67 x +70}}+\frac {\left (\frac {558049940 A}{239274151943121}+\frac {10189562168 B}{239274151943121}+\frac {57928013546 C}{239274151943121}\right ) \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{1311 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}+\frac {\left (\frac {973598080 A}{239274151943121}-\frac {1671781148 B}{239274151943121}-\frac {13468881836 C}{239274151943121}\right ) \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \left (-\frac {23 \operatorname {EllipticE}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{3}+7 \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )\right )}{1311 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}\)	\(276\)
risch	\(-\frac {2 \left (17524765440 x^{5} A -30092060664 B \,x^{5}-242439873048 x^{5} C -319649088360 x^{4} A [phone] x^{4} B [phone] C \,x^{4}[phone] x^{3} A -747876127330 B \,x^{3}-11148381081970 C \,x^{3}-2752459952046 A \,x^{2}-1542113387556 B \,x^{2}[phone] C \,x^{2}-2084092866018 A x [phone] B x [phone] C x [phone] A [phone] B [phone] C \right )}{717822455829363 \left (6 x^{3}-53 x^{2}+67 x +70\right )^{\frac {3}{2}}}+\frac {2 \left (486799040 A -835890574 B -6734440918 C \right ) \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \left (-\frac {23 \operatorname {EllipticE}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{3}+7 \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )\right )}{313688413197431631 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}+\frac {558049940 A \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{313688413197431631 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}+\frac {10189562168 B \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{313688413197431631 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}+\frac {57928013546 C \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{313688413197431631 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}\)	\(368\)
default	\(A \left (\frac {\left (-\frac {3206}{139216401} x^{2}+\frac {3314}{21981537} x -\frac {13687}{835298406}\right ) \sqrt {6 x^{3}-53 x^{2}+67 x +70}}{\left (x^{3}-\frac {53}{6} x^{2}+\frac {67}{6} x +\frac {35}{3}\right )^{2}}-\frac {12 \left (\frac {13336565795}{2153467367488089}+\frac {486799040}{717822455829363} x^{2}-\frac {723013130}{113340387762531} x \right )}{\sqrt {6 x^{3}-53 x^{2}+67 x +70}}+\frac {558049940 \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{313688413197431631 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}+\frac {973598080 \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \left (-\frac {23 \operatorname {EllipticE}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{3}+7 \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )\right )}{313688413197431631 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}\right )+B \left (\frac {\left (\frac {112210}{417649203}-\frac {7331}{139216401} x^{2}+\frac {10585}{43963074} x \right ) \sqrt {6 x^{3}-53 x^{2}+67 x +70}}{\left (x^{3}-\frac {53}{6} x^{2}+\frac {67}{6} x +\frac {35}{3}\right )^{2}}-\frac {12 \left (\frac {26335676756}{2153467367488089}-\frac {835890574}{717822455829363} x^{2}+\frac {361408261}{113340387762531} x \right )}{\sqrt {6 x^{3}-53 x^{2}+67 x +70}}+\frac {10189562168 \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{313688413197431631 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}-\frac {1671781148 \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \left (-\frac {23 \operatorname {EllipticE}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{3}+7 \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )\right )}{313688413197431631 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}\right )+C \left (\frac {\left (\frac {256585}{417649203}-\frac {31238}{139216401} x^{2}+\frac {37663}{43963074} x \right ) \sqrt {6 x^{3}-53 x^{2}+67 x +70}}{\left (x^{3}-\frac {53}{6} x^{2}+\frac {67}{6} x +\frac {35}{3}\right )^{2}}-\frac {12 \left (\frac {210892425319}{4306934734976178}-\frac {6734440918}{717822455829363} x^{2}+\frac {4819508632}{113340387762531} x \right )}{\sqrt {6 x^{3}-53 x^{2}+67 x +70}}+\frac {57928013546 \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{313688413197431631 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}-\frac {13468881836 \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \left (-\frac {23 \operatorname {EllipticE}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{3}+7 \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )\right )}{313688413197431631 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}\right )\)	\(617\)

Input:

int((C*x^2+B*x+A)/(6*x^3-53*x^2+67*x+70)^(5/2),x,method=_RETURNVERBOSE)
                                                                                    
                                                                                    
 

Output:

((-7331/139216401*B-31238/139216401*C-3206/139216401*A)*x^2+(10585/4396307 
4*B+37663/43963074*C+3314/21981537*A)*x+112210/417649203*B+256585/41764920 
3*C-13687/835298406*A)*(6*x^3-53*x^2+67*x+70)^(1/2)/(x^3-53/6*x^2+67/6*x+3 
5/3)^2-12*((486799040/717822455829363*A-835890574/717822455829363*B-673444 
0918/717822455829363*C)*x^2+(-723013130/113340387762531*A[phone]/113340 
387762531*B[phone]/113340387762531*C)*x[phone]/2153467367488089*A 
[phone]/2153467367488089*B[phone]/4306934734976178*C)/(6*x^3-53 
*x^2+67*x+70)^(1/2)+1/1311*(558049940/239274151943121*A[phone]/239274 
151943121*B[phone]/239274151943121*C)*(76+114*x)^(1/2)*(483-69*x)^(1/ 
2)*(285-114*x)^(1/2)/(6*x^3-53*x^2+67*x+70)^(1/2)*EllipticF(1/19*(76+114*x 
)^(1/2),1/46*874^(1/2))+1/1311*(973598080/239274151943121*A-1671781148/239 
274151943121*B-13468881836/239274151943121*C)*(76+114*x)^(1/2)*(483-69*x)^ 
(1/2)*(285-114*x)^(1/2)/(6*x^3-53*x^2+67*x+70)^(1/2)*(-23/3*EllipticE(1/19 
*(76+114*x)^(1/2),1/46*874^(1/2))+7*EllipticF(1/19*(76+114*x)^(1/2),1/46*8 
74^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 358, normalized size of antiderivative = 0.75 \[ \int \frac {A+B x+C x^2}{\left (70+67 x-53 x^2+6 x^3\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

integrate((C*x^2+B*x+A)/(6*x^3-53*x^2+67*x+70)^(5/2),x, algorithm="fricas" 
)
 

Output:

2/6460402102464267*(35*sqrt(6)*(36*(440325694*A + 677197987*B + 2348953618 
*C)*x^6 - 636*(440325694*A + 677197987*B + 2348953618*C)*x^5 + 3613*(44032 
5694*A + 677197987*B + 2348953618*C)*x^4 - 6262*(440325694*A + 677197987*B 
 + 2348953618*C)*x^3 - 2931*(440325694*A + 677197987*B + 2348953618*C)*x^2 
 + 9380*(440325694*A + 677197987*B + 2348953618*C)*x + 2157595900600*A + 3 
318270136300*B + 11509872728200*C)*weierstrassPInverse(1603/27, 18980/729, 
 x - 53/18) - 18*sqrt(6)*(36*(243399520*A - 417945287*B - 3367220459*C)*x^ 
6 - 636*(243399520*A - 417945287*B - 3367220459*C)*x^5 + 3613*(243399520*A 
 - 417945287*B - 3367220459*C)*x^4 - 6262*(243399520*A - 417945287*B - 336 
7220459*C)*x^3 - 2931*(243399520*A - 417945287*B - 3367220459*C)*x^2 + 938 
0*(243399520*A - 417945287*B - 3367220459*C)*x + 1192657648000*A - 2047931 
906300*B - 16499380249100*C)*weierstrassZeta(1603/27, 18980/729, weierstra 
ssPInverse(1603/27, 18980/729, x - 53/18)) - 9*(72*(243399520*A - 41794528 
7*B - 3367220459*C)*x^5 - 60*(5327484806*A - 5803571434*B - 54006669667*C) 
*x^4 + 10*(181188044744*A - 74787612733*B - 1114838108197*C)*x^3 - 21*(131 
069521526*A + 73433970836*B - 55425390757*C)*x^2 - 9*(231565874002*A - 153 
264610481*B - 1764497903840*C)*x + 2078836420243*A + 215556444460*B + 6824 
505272200*C)*sqrt(6*x^3 - 53*x^2 + 67*x + 70))/(36*x^6 - 636*x^5 + 3613*x^ 
4 - 6262*x^3 - 2931*x^2 + 9380*x + 4900)
 

Sympy [F]

\[ \int \frac {A+B x+C x^2}{\left (70+67 x-53 x^2+6 x^3\right )^{5/2}} \, dx=\int \frac {A + B x + C x^{2}}{\left (\left (x - 7\right ) \left (2 x - 5\right ) \left (3 x + 2\right )\right )^{\frac {5}{2}}}\, dx \] Input:

integrate((C*x**2+B*x+A)/(6*x**3-53*x**2+67*x+70)**(5/2),x)
 

Output:

Integral((A + B*x + C*x**2)/((x - 7)*(2*x - 5)*(3*x + 2))**(5/2), x)
 

Maxima [F]

\[ \int \frac {A+B x+C x^2}{\left (70+67 x-53 x^2+6 x^3\right )^{5/2}} \, dx=\int { \frac {C x^{2} + B x + A}{{\left (6 \, x^{3} - 53 \, x^{2} + 67 \, x + 70\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((C*x^2+B*x+A)/(6*x^3-53*x^2+67*x+70)^(5/2),x, algorithm="maxima" 
)
 

Output:

integrate((C*x^2 + B*x + A)/(6*x^3 - 53*x^2 + 67*x + 70)^(5/2), x)
 

Giac [F]

\[ \int \frac {A+B x+C x^2}{\left (70+67 x-53 x^2+6 x^3\right )^{5/2}} \, dx=\int { \frac {C x^{2} + B x + A}{{\left (6 \, x^{3} - 53 \, x^{2} + 67 \, x + 70\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((C*x^2+B*x+A)/(6*x^3-53*x^2+67*x+70)^(5/2),x, algorithm="giac")
 

Output:

integrate((C*x^2 + B*x + A)/(6*x^3 - 53*x^2 + 67*x + 70)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2}{\left (70+67 x-53 x^2+6 x^3\right )^{5/2}} \, dx=\int \frac {C\,x^2+B\,x+A}{{\left (6\,x^3-53\,x^2+67\,x+70\right )}^{5/2}} \,d x \] Input:

int((A + B*x + C*x^2)/(67*x - 53*x^2 + 6*x^3 + 70)^(5/2),x)
 

Output:

int((A + B*x + C*x^2)/(67*x - 53*x^2 + 6*x^3 + 70)^(5/2), x)
 

Reduce [F]

\[ \int \frac {A+B x+C x^2}{\left (70+67 x-53 x^2+6 x^3\right )^{5/2}} \, dx=\text {too large to display} \] Input:

int((C*x^2+B*x+A)/(6*x^3-53*x^2+67*x+70)^(5/2),x)
 

Output:

(2*sqrt(6*x**3 - 53*x**2 + 67*x + 70)*b + 11448*int(sqrt(6*x**3 - 53*x**2 
+ 67*x + 70)/(216*x**9 - 5724*x**8 + 57798*x**7 - 269153*x**6 + 511851*x** 
5 + 44979*x**4 - 1102457*x**3 + 163590*x**2 + 984900*x + 343000),x)*a*x**6 
 - 202248*int(sqrt(6*x**3 - 53*x**2 + 67*x + 70)/(216*x**9 - 5724*x**8 + 5 
7798*x**7 - 269153*x**6 + 511851*x**5 + 44979*x**4 - 1102457*x**3 + 163590 
*x**2 + 984900*x + 343000),x)*a*x**5 + 1148934*int(sqrt(6*x**3 - 53*x**2 + 
 67*x + 70)/(216*x**9 - 5724*x**8 + 57798*x**7 - 269153*x**6 + 511851*x**5 
 + 44979*x**4 - 1102457*x**3 + 163590*x**2 + 984900*x + 343000),x)*a*x**4 
- 1991316*int(sqrt(6*x**3 - 53*x**2 + 67*x + 70)/(216*x**9 - 5724*x**8 + 5 
7798*x**7 - 269153*x**6 + 511851*x**5 + 44979*x**4 - 1102457*x**3 + 163590 
*x**2 + 984900*x + 343000),x)*a*x**3 - 932058*int(sqrt(6*x**3 - 53*x**2 + 
67*x + 70)/(216*x**9 - 5724*x**8 + 57798*x**7 - 269153*x**6 + 511851*x**5 
+ 44979*x**4 - 1102457*x**3 + 163590*x**2 + 984900*x + 343000),x)*a*x**2 + 
 2982840*int(sqrt(6*x**3 - 53*x**2 + 67*x + 70)/(216*x**9 - 5724*x**8 + 57 
798*x**7 - 269153*x**6 + 511851*x**5 + 44979*x**4 - 1102457*x**3 + 163590* 
x**2 + 984900*x + 343000),x)*a*x + 1558200*int(sqrt(6*x**3 - 53*x**2 + 67* 
x + 70)/(216*x**9 - 5724*x**8 + 57798*x**7 - 269153*x**6 + 511851*x**5 + 4 
4979*x**4 - 1102457*x**3 + 163590*x**2 + 984900*x + 343000),x)*a + 7236*in 
t(sqrt(6*x**3 - 53*x**2 + 67*x + 70)/(216*x**9 - 5724*x**8 + 57798*x**7 - 
269153*x**6 + 511851*x**5 + 44979*x**4 - 1102457*x**3 + 163590*x**2 + 9...