\(\int (A+B x+C x^2) (70+67 x-53 x^2+6 x^3)^{3/2} \, dx\) [118]

Optimal result

Integrand size = 30, antiderivative size = 498 \[ \int \left (A+B x+C x^2\right ) \left (70+67 x-53 x^2+6 x^3\right )^{3/2} \, dx=-\frac {(5429970 A-13817781 B-101582932 C) \left (70+67 x-53 x^2+6 x^3\right )^{3/2}}{1990170 (5-2 x) (7-x)}+\frac {(1687014 A+186462072 B+1091774983 C) \left (70+67 x-53 x^2+6 x^3\right )^{3/2}}{673596 (5-2 x) (7-x) (2+3 x)}-\frac {1}{351} (9 B+53 C) (5-2 x) (2+3 x) \left (70+67 x-53 x^2+6 x^3\right )^{3/2}+\frac {(51246 A+36513 B+24272 C) (2+3 x) \left (70+67 x-53 x^2+6 x^3\right )^{3/2}}{104247 (7-x)}+\frac {(3970512 A+4397985 B+11120117 C) (2+3 x) \left (70+67 x-53 x^2+6 x^3\right )^{3/2}}{2189187 (5-2 x) (7-x)}+\frac {(702 A-342 B-4627 C) (5-2 x) (2+3 x) \left (70+67 x-53 x^2+6 x^3\right )^{3/2}}{23166 (7-x)}+\frac {1}{45} C (5-2 x) (7-x) (2+3 x) \left (70+67 x-53 x^2+6 x^3\right )^{3/2}-\frac {\sqrt {19} (85433231520 A+588536034399 B+3147821840803 C) \left (70+67 x-53 x^2+6 x^3\right )^{3/2} E\left (\arcsin \left (\frac {\sqrt {2+3 x}}{\sqrt {23}}\right )|\frac {46}{19}\right )}{131351220 (5-2 x)^{3/2} (7-x)^{3/2} (2+3 x)^{3/2}}-\frac {\sqrt {19} (3070783170 A+11473156917 B+56134008934 C) \left (70+67 x-53 x^2+6 x^3\right )^{3/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {19}{2}}}{\sqrt {2+3 x}}\right ),\frac {46}{19}\right )}{4864860 (-7+x)^{3/2} (-5+2 x)^{3/2} (2+3 x)^{3/2}} \] Output:

-1/1990170*(5429970*A-13817781*B-101582932*C)*(6*x^3-53*x^2+67*x+70)^(3/2) 
/(5-2*x)/(7-x)+1/673596*(1687014*A[phone]*B[phone]*C)*(6*x^3-53*x^2 
+67*x+70)^(3/2)/(5-2*x)/(7-x)/(2+3*x)-1/351*(9*B+53*C)*(5-2*x)*(2+3*x)*(6* 
x^3-53*x^2+67*x+70)^(3/2)+(51246*A+36513*B+24272*C)*(2+3*x)*(6*x^3-53*x^2+ 
67*x+70)^(3/2)/(729729-104247*x)+1/2189187*(3970512*A+4397985*B[phone]*C 
)*(2+3*x)*(6*x^3-53*x^2+67*x+70)^(3/2)/(5-2*x)/(7-x)+(702*A-342*B-4627*C)* 
(5-2*x)*(2+3*x)*(6*x^3-53*x^2+67*x+70)^(3/2)/(162162-23166*x)+1/45*C*(5-2* 
x)*(7-x)*(2+3*x)*(6*x^3-53*x^2+67*x+70)^(3/2)-1/131351220*19^(1/2)*(854332 
31520*A[phone]*B[phone]*C)*(6*x^3-53*x^2+67*x+70)^(3/2)*Ellip 
ticE(1/23*(2+3*x)^(1/2)*23^(1/2),1/19*874^(1/2))/(5-2*x)^(3/2)/(7-x)^(3/2) 
/(2+3*x)^(3/2)-1/4864860*19^(1/2)*(3070783170*A[phone]*B[phone]* 
C)*(6*x^3-53*x^2+67*x+70)^(3/2)*EllipticF(1/2*38^(1/2)/(2+3*x)^(1/2),1/19* 
874^(1/2))/(-7+x)^(3/2)/(-5+2*x)^(3/2)/(2+3*x)^(3/2)
 

Mathematica [A] (verified)

Time = 10.60 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.51 \[ \int \left (A+B x+C x^2\right ) \left (70+67 x-53 x^2+6 x^3\right )^{3/2} \, dx=-\frac {\sqrt {5-2 x} \left (6 \sqrt {5-2 x} \left (-14-19 x+3 x^2\right ) \left (24570 A \left (30587+81630 x+47538 x^2-22896 x^3+1944 x^4\right )+18 B \left (696448058+55894437 x+75120120 x^2+47096910 x^3-25242840 x^4+2245320 x^5\right )+C \left (75793722863+7585816032 x+763363998 x^2+993475476 x^3+661818276 x^4-380806272 x^5+35026992 x^6\right )\right )+2 \sqrt {46} (85433231520 A+588536034399 B+3147821840803 C) \sqrt {7-x} \sqrt {2+3 x} E\left (\arcsin \left (\sqrt {\frac {2}{19}} \sqrt {2+3 x}\right )|\frac {19}{46}\right )-27 \sqrt {46} (6251222250 A+35066348514 B+183233391763 C) \sqrt {7-x} \sqrt {2+3 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{19}} \sqrt {2+3 x}\right ),\frac {19}{46}\right )\right )}{262702440 \sqrt {70+67 x-53 x^2+6 x^3}} \] Input:

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

Output:

-1/262702440*(Sqrt[5 - 2*x]*(6*Sqrt[5 - 2*x]*(-14 - 19*x + 3*x^2)*(24570*A 
*(30587 + 81630*x + 47538*x^2 - 22896*x^3 + 1944*x^4) + 18*B*(696448058 + 
55894437*x + 75120120*x^2 + 47096910*x^3 - 25242840*x^4 + 2245320*x^5) + C 
*(75793722863 + 7585816032*x + 763363998*x^2 + 993475476*x^3 + 661818276*x 
^4 - 380806272*x^5 + 35026992*x^6)) + 2*Sqrt[46]*(85433231520*A + 58853603 
4399*B + 3147821840803*C)*Sqrt[7 - x]*Sqrt[2 + 3*x]*EllipticE[ArcSin[Sqrt[ 
2/19]*Sqrt[2 + 3*x]], 19/46] - 27*Sqrt[46]*(6251222250*A + 35066348514*B + 
 183233391763*C)*Sqrt[7 - x]*Sqrt[2 + 3*x]*EllipticF[ArcSin[Sqrt[2/19]*Sqr 
t[2 + 3*x]], 19/46]))/Sqrt[70 + 67*x - 53*x^2 + 6*x^3]
 

Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 16.15 (sec) , antiderivative size = 2634, normalized size of antiderivative = 5.29, 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, 1236, 27, 1236, 27, 1231, 27, 1231, 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.

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

\(\Big \downarrow \) 2526

\(\displaystyle \frac {1}{18} \int (18 A-67 C+2 (9 B+53 C) x) \left (6 x^3-53 x^2+67 x+70\right )^{3/2}dx+\frac {1}{45} C \left (6 x^3-53 x^2+67 x+70\right )^{5/2}\)

\(\Big \downarrow \) 2490

\(\displaystyle \frac {1}{18} \int \left (\frac {1}{18} (18 (18 A-67 C)+106 (9 B+53 C))+2 (9 B+53 C) \left (x-\frac {53}{18}\right )\right ) \left (6 \left (x-\frac {53}{18}\right )^3-\frac {1603}{18} \left (x-\frac {53}{18}\right )-\frac {9490}{243}\right )^{3/2}d\left (x-\frac {53}{18}\right )+\frac {1}{45} C \left (6 x^3-53 x^2+67 x+70\right )^{5/2}\)

\(\Big \downarrow \) 2486

\(\displaystyle \frac {1}{45} C \left (6 x^3-53 x^2+67 x+70\right )^{5/2}+\frac {\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{3/2} \int \frac {\left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \left (162 A+477 B+2206 C+18 (9 B+53 C) \left (x-\frac {53}{18}\right )\right ) \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{3/2}}{729 \sqrt {3}}d\left (x-\frac {53}{18}\right )}{972 \sqrt {2} \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{45} C \left (6 x^3-53 x^2+67 x+70\right )^{5/2}+\frac {\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{3/2} \int \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \left (162 A+477 B+2206 C+18 (9 B+53 C) \left (x-\frac {53}{18}\right )\right ) \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{3/2}d\left (x-\frac {53}{18}\right )}{708588 \sqrt {6} \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{3/2}}\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {1}{45} C \left (6 x^3-53 x^2+67 x+70\right )^{5/2}+\frac {\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{3/2} \left (\frac {\int 162 \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}}}} \left (\frac {5 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (\left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (9 B+53 C)-2 \sqrt [3]{18980+35397 i \sqrt {3}} (162 A+477 B+2206 C)\right )-3 \left (\left (2569609-1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\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 (18980+35397 i \sqrt {3}\right )^{2/3}}+18 \left (2106 A+\left (6201-\frac {158697}{\sqrt [3]{18980+35397 i \sqrt {3}}}-99 \sqrt [3]{18980+35397 i \sqrt {3}}\right ) B+\left (28678-\frac {934549}{\sqrt [3]{18980+35397 i \sqrt {3}}}-583 \sqrt [3]{18980+35397 i \sqrt {3}}\right ) C\right ) \left (x-\frac {53}{18}\right )\right ) \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 )}{2106}+\frac {1}{117} \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 )^{5/2} (9 B+53 C)\right )}{708588 \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 )^{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}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{45} C \left (6 x^3-53 x^2+67 x+70\right )^{5/2}+\frac {\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{3/2} \left (\frac {1}{13} \int \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}}}} \left (\frac {5 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (\left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (9 B+53 C)-2 \sqrt [3]{18980+35397 i \sqrt {3}} (162 A+477 B+2206 C)\right )-3 \left (\left (2569609-1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\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 (18980+35397 i \sqrt {3}\right )^{2/3}}+18 \left (2106 A+\left (6201-\frac {158697}{\sqrt [3]{18980+35397 i \sqrt {3}}}-99 \sqrt [3]{18980+35397 i \sqrt {3}}\right ) B+\left (28678-\frac {934549}{\sqrt [3]{18980+35397 i \sqrt {3}}}-583 \sqrt [3]{18980+35397 i \sqrt {3}}\right ) C\right ) \left (x-\frac {53}{18}\right )\right ) \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 )+\frac {1}{117} \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 )^{5/2} (9 B+53 C)\right )}{708588 \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 )^{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}}\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {1}{45} C \left (6 x^3-53 x^2+67 x+70\right )^{5/2}+\frac {\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{3/2} \left (\frac {1}{117} (9 B+53 C) \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 )^{5/2}+\frac {1}{13} \left (\frac {1}{99} \left (2106 A+\left (6201-\frac {158697}{\sqrt [3]{18980+35397 i \sqrt {3}}}-99 \sqrt [3]{18980+35397 i \sqrt {3}}\right ) B+\left (28678-\frac {934549}{\sqrt [3]{18980+35397 i \sqrt {3}}}-583 \sqrt [3]{18980+35397 i \sqrt {3}}\right ) C\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}}}} \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}+\frac {\int \frac {4374 \left (\frac {234 \left (17633 \left (18980 i-35397 \sqrt {3}\right )[phone] i \sqrt [3]{18980+35397 i \sqrt {3}}+5 \left (18980 i-35397 \sqrt {3}\right ) \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) A+9 \left (1257113 \left (18980 i-35397 \sqrt {3}\right )+456855 \sqrt [3]{18980+35397 i \sqrt {3}} \left (1339 i+1518 \sqrt {3}\right )-285 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (84706 i+47541 \sqrt {3}\right )\right ) B+2 \left (25634323 \left (18980 i-35397 \sqrt {3}\right )+8015 \sqrt [3]{18980+35397 i \sqrt {3}} \left (1324453 i+2292939 \sqrt {3}\right )-5 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (136214203 i+56395287 \sqrt {3}\right )\right ) C}{18980 i-35397 \sqrt {3}}+\frac {18 \left (1404 i \left (18980 i-35397 \sqrt {3}+1603 i \sqrt [3]{18980+35397 i \sqrt {3}}\right ) A+9 \left (703753-16259022 i \sqrt {3}+17633 \left (18980+35397 i \sqrt {3}\right )^{2/3}-\sqrt [3]{18980+35397 i \sqrt {3}} \left (666718-129789 i \sqrt {3}\right )\right ) B+\left (136488389-676743444 i \sqrt {3}+934549 \left (18980+35397 i \sqrt {3}\right )^{2/3}-\sqrt [3]{18980+35397 i \sqrt {3}} \left (26958776-6878817 i \sqrt {3}\right )\right ) C\right ) \left (x-\frac {53}{18}\right )}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right ) \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}}{\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}}}}}d\left (x-\frac {53}{18}\right )}{1782}\right )\right )}{708588 \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 )^{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}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{45} C \left (6 x^3-53 x^2+67 x+70\right )^{5/2}+\frac {\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{3/2} \left (\frac {1}{117} (9 B+53 C) \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 )^{5/2}+\frac {1}{13} \left (\frac {1}{99} \left (2106 A+\left (6201-\frac {158697}{\sqrt [3]{18980+35397 i \sqrt {3}}}-99 \sqrt [3]{18980+35397 i \sqrt {3}}\right ) B+\left (28678-\frac {934549}{\sqrt [3]{18980+35397 i \sqrt {3}}}-583 \sqrt [3]{18980+35397 i \sqrt {3}}\right ) C\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}}}} \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}+\frac {27}{11} \int \frac {\left (\frac {234 \left (17633 \left (18980 i-35397 \sqrt {3}\right )[phone] i \sqrt [3]{18980+35397 i \sqrt {3}}+5 \left (18980 i-35397 \sqrt {3}\right ) \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) A+9 \left (1257113 \left (18980 i-35397 \sqrt {3}\right )+456855 \sqrt [3]{18980+35397 i \sqrt {3}} \left (1339 i+1518 \sqrt {3}\right )-285 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (84706 i+47541 \sqrt {3}\right )\right ) B+2 \left (25634323 \left (18980 i-35397 \sqrt {3}\right )+8015 \sqrt [3]{18980+35397 i \sqrt {3}} \left (1324453 i+2292939 \sqrt {3}\right )-5 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (136214203 i+56395287 \sqrt {3}\right )\right ) C}{18980 i-35397 \sqrt {3}}+\frac {18 \left (1404 i \left (18980 i-35397 \sqrt {3}+1603 i \sqrt [3]{18980+35397 i \sqrt {3}}\right ) A+9 \left (703753-16259022 i \sqrt {3}+17633 \left (18980+35397 i \sqrt {3}\right )^{2/3}-\sqrt [3]{18980+35397 i \sqrt {3}} \left (666718-129789 i \sqrt {3}\right )\right ) B+\left (136488389-676743444 i \sqrt {3}+934549 \left (18980+35397 i \sqrt {3}\right )^{2/3}-\sqrt [3]{18980+35397 i \sqrt {3}} \left (26958776-6878817 i \sqrt {3}\right )\right ) C\right ) \left (x-\frac {53}{18}\right )}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right ) \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}}{\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}}}}}d\left (x-\frac {53}{18}\right )\right )\right )}{708588 \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 )^{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}}\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {1}{45} C \left (6 x^3-53 x^2+67 x+70\right )^{5/2}+\frac {\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{3/2} \left (\frac {1}{117} (9 B+53 C) \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 )^{5/2}+\frac {1}{13} \left (\frac {1}{99} \left (2106 A+\left (6201-\frac {158697}{\sqrt [3]{18980+35397 i \sqrt {3}}}-99 \sqrt [3]{18980+35397 i \sqrt {3}}\right ) B+\left (28678-\frac {934549}{\sqrt [3]{18980+35397 i \sqrt {3}}}-583 \sqrt [3]{18980+35397 i \sqrt {3}}\right ) C\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}}}} \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}+\frac {27}{11} \left (-\frac {1}{567} \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}}}} \left (\frac {4914 \left (2519 \left (18980 i-35397 \sqrt {3}\right )+2569609 i \sqrt [3]{18980+35397 i \sqrt {3}}+\left (18980 i-35397 \sqrt {3}\right ) \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) A+9 \left (3353779 \left (18980 i-35397 \sqrt {3}\right )+1918791 \sqrt [3]{18980+35397 i \sqrt {3}} \left (1339 i+1518 \sqrt {3}\right )-1197 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (84706 i+47541 \sqrt {3}\right )\right ) B+2 \left (65837629 \left (18980 i-35397 \sqrt {3}\right )+33663 \sqrt [3]{18980+35397 i \sqrt {3}} \left (1324453 i+2292939 \sqrt {3}\right )-21 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (136214203 i+56395287 \sqrt {3}\right )\right ) C}{18980 i-35397 \sqrt {3}}-\frac {126 \left (1404 i \left (18980 i-35397 \sqrt {3}+1603 i \sqrt [3]{18980+35397 i \sqrt {3}}\right ) A+9 \left (703753-16259022 i \sqrt {3}+17633 \left (18980+35397 i \sqrt {3}\right )^{2/3}-\sqrt [3]{18980+35397 i \sqrt {3}} \left (666718-129789 i \sqrt {3}\right )\right ) B+\left (136488389-676743444 i \sqrt {3}+934549 \left (18980+35397 i \sqrt {3}\right )^{2/3}-\sqrt [3]{18980+35397 i \sqrt {3}} \left (26958776-6878817 i \sqrt {3}\right )\right ) C\right ) \left (x-\frac {53}{18}\right )}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right ) \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}-\frac {\int -\frac {314928 \left (\frac {4914 \left (67262372984525294-3380595733652293260 i \sqrt {3}+2569609 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (3398602427-1343670120 i \sqrt {3}\right )+458 i \sqrt [3]{18980+35397 i \sqrt {3}} \left (18578011448230 i+184057477302519 \sqrt {3}\right )\right ) A-9 \left (12341275759119521070766+12167647667724220303149 i \sqrt {3}-4911981529 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (3398602427-1343670120 i \sqrt {3}\right )+\sqrt [3]{18980+35397 i \sqrt {3}} \left (344151862300448490851-290775169602083996622 i \sqrt {3}\right )\right ) B-\left (655317911297594568903152+583050849825149580048237 i \sqrt {3}-213334302818 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (3398602427-1343670120 i \sqrt {3}\right )+\sqrt [3]{18980+35397 i \sqrt {3}} \left (18084415535334764697163-13869183333568560057684 i \sqrt {3}\right )\right ) C}{\left (18980+35397 i \sqrt {3}\right )^{8/3}}+\frac {18 \left (24570 \left (367087-3796 \sqrt [3]{18980+35397 i \sqrt {3}}+229 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) A+9 \left (3508558235-96466811 \sqrt [3]{18980+35397 i \sqrt {3}}+2188745 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) B+\left (152381644870-4765577803 \sqrt [3]{18980+35397 i \sqrt {3}}[phone] \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 ) \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}}{\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}}}}}d\left (x-\frac {53}{18}\right )}{1102248}\right )\right )\right )}{708588 \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 )^{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}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{45} C \left (6 x^3-53 x^2+67 x+70\right )^{5/2}+\frac {\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{3/2} \left (\frac {1}{117} (9 B+53 C) \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 )^{5/2}+\frac {1}{13} \left (\frac {1}{99} \left (2106 A+\left (6201-\frac {158697}{\sqrt [3]{18980+35397 i \sqrt {3}}}-99 \sqrt [3]{18980+35397 i \sqrt {3}}\right ) B+\left (28678-\frac {934549}{\sqrt [3]{18980+35397 i \sqrt {3}}}-583 \sqrt [3]{18980+35397 i \sqrt {3}}\right ) C\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}}}} \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}+\frac {27}{11} \left (\frac {2}{7} \int \frac {\left (\frac {4914 \left (67262372984525294-3380595733652293260 i \sqrt {3}+2569609 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (3398602427-1343670120 i \sqrt {3}\right )+458 i \sqrt [3]{18980+35397 i \sqrt {3}} \left (18578011448230 i+184057477302519 \sqrt {3}\right )\right ) A-9 \left (12341275759119521070766+12167647667724220303149 i \sqrt {3}-4911981529 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (3398602427-1343670120 i \sqrt {3}\right )+\sqrt [3]{18980+35397 i \sqrt {3}} \left (344151862300448490851-290775169602083996622 i \sqrt {3}\right )\right ) B-\left (655317911297594568903152+583050849825149580048237 i \sqrt {3}-213334302818 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (3398602427-1343670120 i \sqrt {3}\right )+\sqrt [3]{18980+35397 i \sqrt {3}} \left (18084415535334764697163-13869183333568560057684 i \sqrt {3}\right )\right ) C}{\left (18980+35397 i \sqrt {3}\right )^{8/3}}+\frac {18 \left (24570 \left (367087-3796 \sqrt [3]{18980+35397 i \sqrt {3}}+229 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) A+9 \left (3508558235-96466811 \sqrt [3]{18980+35397 i \sqrt {3}}+2188745 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) B+\left (152381644870-4765577803 \sqrt [3]{18980+35397 i \sqrt {3}}[phone] \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 ) \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}}{\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}}}}}d\left (x-\frac {53}{18}\right )-\frac {1}{567} \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}}}} \left (\frac {4914 \left (2519 \left (18980 i-35397 \sqrt {3}\right )+2569609 i \sqrt [3]{18980+35397 i \sqrt {3}}+\left (18980 i-35397 \sqrt {3}\right ) \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) A+9 \left (3353779 \left (18980 i-35397 \sqrt {3}\right )+1918791 \sqrt [3]{18980+35397 i \sqrt {3}} \left (1339 i+1518 \sqrt {3}\right )-1197 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (84706 i+47541 \sqrt {3}\right )\right ) B+2 \left (65837629 \left (18980 i-35397 \sqrt {3}\right )+33663 \sqrt [3]{18980+35397 i \sqrt {3}} \left (1324453 i+2292939 \sqrt {3}\right )-21 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (136214203 i+56395287 \sqrt {3}\right )\right ) C}{18980 i-35397 \sqrt {3}}-\frac {126 \left (1404 i \left (18980 i-35397 \sqrt {3}+1603 i \sqrt [3]{18980+35397 i \sqrt {3}}\right ) A+9 \left (703753-16259022 i \sqrt {3}+17633 \left (18980+35397 i \sqrt {3}\right )^{2/3}-\sqrt [3]{18980+35397 i \sqrt {3}} \left (666718-129789 i \sqrt {3}\right )\right ) B+\left (136488389-676743444 i \sqrt {3}+934549 \left (18980+35397 i \sqrt {3}\right )^{2/3}-\sqrt [3]{18980+35397 i \sqrt {3}} \left (26958776-6878817 i \sqrt {3}\right )\right ) C\right ) \left (x-\frac {53}{18}\right )}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right ) \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 )\right )\right )}{708588 \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 )^{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}}\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {1}{45} C \left (6 x^3-53 x^2+67 x+70\right )^{5/2}+\frac {\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{3/2} \left (\frac {1}{117} (9 B+53 C) \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 )^{5/2}+\frac {1}{13} \left (\frac {1}{99} \left (2106 A+\left (6201-\frac {158697}{\sqrt [3]{18980+35397 i \sqrt {3}}}-99 \sqrt [3]{18980+35397 i \sqrt {3}}\right ) B+\left (28678-\frac {934549}{\sqrt [3]{18980+35397 i \sqrt {3}}}-583 \sqrt [3]{18980+35397 i \sqrt {3}}\right ) C\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}}}} \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}+\frac {27}{11} \left (\frac {2}{7} \left (-\frac {1}{45} \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} \left (\frac {65 \left (2569609 \left (3398602427-1343670120 i \sqrt {3}\right )+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (3398602427-1343670120 i \sqrt {3}\right )+\sqrt [3]{18980+35397 i \sqrt {3}} \left (207191147777380+94797471230919 i \sqrt {3}\right )\right ) (86562 A+303057 B+1462466 C)}{\left (18980+35397 i \sqrt {3}\right )^{8/3}}-\frac {18 \left (24570 \left (367087-3796 \sqrt [3]{18980+35397 i \sqrt {3}}+229 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) A+9 \left (3508558235-96466811 \sqrt [3]{18980+35397 i \sqrt {3}}+2188745 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) B+\left (152381644870-4765577803 \sqrt [3]{18980+35397 i \sqrt {3}}[phone] \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 )-\frac {\int -\frac {2834352 \left (455 (11888793738 A+49154894001 B+245214976870 C)[phone] A+588536034399 B+3147821840803 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 )}{787320}\right )-\frac {1}{567} \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}}}} \left (\frac {4914 \left (2519 \left (18980 i-35397 \sqrt {3}\right )+2569609 i \sqrt [3]{18980+35397 i \sqrt {3}}+\left (18980 i-35397 \sqrt {3}\right ) \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) A+9 \left (3353779 \left (18980 i-35397 \sqrt {3}\right )+1918791 \sqrt [3]{18980+35397 i \sqrt {3}} \left (1339 i+1518 \sqrt {3}\right )-1197 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (84706 i+47541 \sqrt {3}\right )\right ) B+2 \left (65837629 \left (18980 i-35397 \sqrt {3}\right )+33663 \sqrt [3]{18980+35397 i \sqrt {3}} \left (1324453 i+2292939 \sqrt {3}\right )-21 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (136214203 i+56395287 \sqrt {3}\right )\right ) C}{18980 i-35397 \sqrt {3}}-\frac {126 \left (1404 i \left (18980 i-35397 \sqrt {3}+1603 i \sqrt [3]{18980+35397 i \sqrt {3}}\right ) A+9 \left (703753-16259022 i \sqrt {3}+17633 \left (18980+35397 i \sqrt {3}\right )^{2/3}-\sqrt [3]{18980+35397 i \sqrt {3}} \left (666718-129789 i \sqrt {3}\right )\right ) B+\left (136488389-676743444 i \sqrt {3}+934549 \left (18980+35397 i \sqrt {3}\right )^{2/3}-\sqrt [3]{18980+35397 i \sqrt {3}} \left (26958776-6878817 i \sqrt {3}\right )\right ) C\right ) \left (x-\frac {53}{18}\right )}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right ) \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 )\right )\right )}{708588 \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 )^{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}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{45} C \left (6 x^3-53 x^2+67 x+70\right )^{5/2}+\frac {\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{3/2} \left (\frac {1}{117} (9 B+53 C) \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 )^{5/2}+\frac {1}{13} \left (\frac {1}{99} \left (2106 A+\left (6201-\frac {158697}{\sqrt [3]{18980+35397 i \sqrt {3}}}-99 \sqrt [3]{18980+35397 i \sqrt {3}}\right ) B+\left (28678-\frac {934549}{\sqrt [3]{18980+35397 i \sqrt {3}}}-583 \sqrt [3]{18980+35397 i \sqrt {3}}\right ) C\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}}}} \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}+\frac {27}{11} \left (\frac {2}{7} \left (\frac {18}{5} \int \frac {455 (11888793738 A+49154894001 B+245214976870 C)[phone] A+588536034399 B+3147821840803 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 )-\frac {1}{45} \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}}}} \left (\frac {65 \left (2569609 \left (3398602427-1343670120 i \sqrt {3}\right )+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (3398602427-1343670120 i \sqrt {3}\right )+\sqrt [3]{18980+35397 i \sqrt {3}} \left (207191147777380+94797471230919 i \sqrt {3}\right )\right ) (86562 A+303057 B+1462466 C)}{\left (18980+35397 i \sqrt {3}\right )^{8/3}}-\frac {18 \left (24570 \left (367087-3796 \sqrt [3]{18980+35397 i \sqrt {3}}+229 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) A+9 \left (3508558235-96466811 \sqrt [3]{18980+35397 i \sqrt {3}}+2188745 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) B+\left (152381644870-4765577803 \sqrt [3]{18980+35397 i \sqrt {3}}[phone] \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 ) \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 )-\frac {1}{567} \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}}}} \left (\frac {4914 \left (2519 \left (18980 i-35397 \sqrt {3}\right )+2569609 i \sqrt [3]{18980+35397 i \sqrt {3}}+\left (18980 i-35397 \sqrt {3}\right ) \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) A+9 \left (3353779 \left (18980 i-35397 \sqrt {3}\right )+1918791 \sqrt [3]{18980+35397 i \sqrt {3}} \left (1339 i+1518 \sqrt {3}\right )-1197 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (84706 i+47541 \sqrt {3}\right )\right ) B+2 \left (65837629 \left (18980 i-35397 \sqrt {3}\right )+33663 \sqrt [3]{18980+35397 i \sqrt {3}} \left (1324453 i+2292939 \sqrt {3}\right )-21 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (136214203 i+56395287 \sqrt {3}\right )\right ) C}{18980 i-35397 \sqrt {3}}-\frac {126 \left (1404 i \left (18980 i-35397 \sqrt {3}+1603 i \sqrt [3]{18980+35397 i \sqrt {3}}\right ) A+9 \left (703753-16259022 i \sqrt {3}+17633 \left (18980+35397 i \sqrt {3}\right )^{2/3}-\sqrt [3]{18980+35397 i \sqrt {3}} \left (666718-129789 i \sqrt {3}\right )\right ) B+\left (136488389-676743444 i \sqrt {3}+934549 \left (18980+35397 i \sqrt {3}\right )^{2/3}-\sqrt [3]{18980+35397 i \sqrt {3}} \left (26958776-6878817 i \sqrt {3}\right )\right ) C\right ) \left (x-\frac {53}{18}\right )}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right ) \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 )\right )\right )}{708588 \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 )^{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}}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {1}{45} C \left (6 x^3-53 x^2+67 x+70\right )^{5/2}+\frac {\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{3/2} \left (\frac {1}{117} (9 B+53 C) \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 )^{5/2}+\frac {1}{13} \left (\frac {1}{99} \left (2106 A+\left (6201-\frac {158697}{\sqrt [3]{18980+35397 i \sqrt {3}}}-99 \sqrt [3]{18980+35397 i \sqrt {3}}\right ) B+\left (28678-\frac {934549}{\sqrt [3]{18980+35397 i \sqrt {3}}}-583 \sqrt [3]{18980+35397 i \sqrt {3}}\right ) C\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}}}} \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}+\frac {27}{11} \left (\frac {2}{7} \left (\frac {18}{5} \left (\left (455 (11888793738 A+49154894001 B+245214976870 C)+\frac {\left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (85433231520 A+588536034399 B+3147821840803 C)}{\sqrt [3]{18980+35397 i \sqrt {3}}}\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 )+(85433231520 A+588536034399 B+3147821840803 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 )-\frac {1}{45} \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}}}} \left (\frac {65 \left (2569609 \left (3398602427-1343670120 i \sqrt {3}\right )+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (3398602427-1343670120 i \sqrt {3}\right )+\sqrt [3]{18980+35397 i \sqrt {3}} \left (207191147777380+94797471230919 i \sqrt {3}\right )\right ) (86562 A+303057 B+1462466 C)}{\left (18980+35397 i \sqrt {3}\right )^{8/3}}-\frac {18 \left (24570 \left (367087-3796 \sqrt [3]{18980+35397 i \sqrt {3}}+229 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) A+9 \left (3508558235-96466811 \sqrt [3]{18980+35397 i \sqrt {3}}+2188745 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) B+\left (152381644870-4765577803 \sqrt [3]{18980+35397 i \sqrt {3}}[phone] \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 ) \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 )-\frac {1}{567} \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}}}} \left (\frac {4914 \left (2519 \left (18980 i-35397 \sqrt {3}\right )+2569609 i \sqrt [3]{18980+35397 i \sqrt {3}}+\left (18980 i-35397 \sqrt {3}\right ) \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) A+9 \left (3353779 \left (18980 i-35397 \sqrt {3}\right )+1918791 \sqrt [3]{18980+35397 i \sqrt {3}} \left (1339 i+1518 \sqrt {3}\right )-1197 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (84706 i+47541 \sqrt {3}\right )\right ) B+2 \left (65837629 \left (18980 i-35397 \sqrt {3}\right )+33663 \sqrt [3]{18980+35397 i \sqrt {3}} \left (1324453 i+2292939 \sqrt {3}\right )-21 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (136214203 i+56395287 \sqrt {3}\right )\right ) C}{18980 i-35397 \sqrt {3}}-\frac {126 \left (1404 i \left (18980 i-35397 \sqrt {3}+1603 i \sqrt [3]{18980+35397 i \sqrt {3}}\right ) A+9 \left (703753-16259022 i \sqrt {3}+17633 \left (18980+35397 i \sqrt {3}\right )^{2/3}-\sqrt [3]{18980+35397 i \sqrt {3}} \left (666718-129789 i \sqrt {3}\right )\right ) B+\left (136488389-676743444 i \sqrt {3}+934549 \left (18980+35397 i \sqrt {3}\right )^{2/3}-\sqrt [3]{18980+35397 i \sqrt {3}} \left (26958776-6878817 i \sqrt {3}\right )\right ) C\right ) \left (x-\frac {53}{18}\right )}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right ) \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 )\right )\right )}{708588 \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 )^{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}}\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {1}{45} C \left (6 x^3-53 x^2+67 x+70\right )^{5/2}+\frac {\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{3/2} \left (\frac {1}{117} (9 B+53 C) \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 )^{5/2}+\frac {1}{13} \left (\frac {1}{99} \left (2106 A+\left (6201-\frac {158697}{\sqrt [3]{18980+35397 i \sqrt {3}}}-99 \sqrt [3]{18980+35397 i \sqrt {3}}\right ) B+\left (28678-\frac {934549}{\sqrt [3]{18980+35397 i \sqrt {3}}}-583 \sqrt [3]{18980+35397 i \sqrt {3}}\right ) C\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}}}} \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}+\frac {27}{11} \left (\frac {2}{7} \left (\frac {18}{5} \left (-\frac {\sqrt {\frac {2}{3}} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (455 (11888793738 A+49154894001 B+245214976870 C)+\frac {\left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (85433231520 A+588536034399 B+3147821840803 C)}{\sqrt [3]{18980+35397 i \sqrt {3}}}\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 ) (85433231520 A+588536034399 B+3147821840803 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 )-\frac {1}{45} \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}}}} \left (\frac {65 \left (2569609 \left (3398602427-1343670120 i \sqrt {3}\right )+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (3398602427-1343670120 i \sqrt {3}\right )+\sqrt [3]{18980+35397 i \sqrt {3}} \left (207191147777380+94797471230919 i \sqrt {3}\right )\right ) (86562 A+303057 B+1462466 C)}{\left (18980+35397 i \sqrt {3}\right )^{8/3}}-\frac {18 \left (24570 \left (367087-3796 \sqrt [3]{18980+35397 i \sqrt {3}}+229 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) A+9 \left (3508558235-96466811 \sqrt [3]{18980+35397 i \sqrt {3}}+2188745 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) B+\left (152381644870-4765577803 \sqrt [3]{18980+35397 i \sqrt {3}}[phone] \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 ) \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 )-\frac {1}{567} \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}}}} \left (\frac {4914 \left (2519 \left (18980 i-35397 \sqrt {3}\right )+2569609 i \sqrt [3]{18980+35397 i \sqrt {3}}+\left (18980 i-35397 \sqrt {3}\right ) \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) A+9 \left (3353779 \left (18980 i-35397 \sqrt {3}\right )+1918791 \sqrt [3]{18980+35397 i \sqrt {3}} \left (1339 i+1518 \sqrt {3}\right )-1197 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (84706 i+47541 \sqrt {3}\right )\right ) B+2 \left (65837629 \left (18980 i-35397 \sqrt {3}\right )+33663 \sqrt [3]{18980+35397 i \sqrt {3}} \left (1324453 i+2292939 \sqrt {3}\right )-21 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (136214203 i+56395287 \sqrt {3}\right )\right ) C}{18980 i-35397 \sqrt {3}}-\frac {126 \left (1404 i \left (18980 i-35397 \sqrt {3}+1603 i \sqrt [3]{18980+35397 i \sqrt {3}}\right ) A+9 \left (703753-16259022 i \sqrt {3}+17633 \left (18980+35397 i \sqrt {3}\right )^{2/3}-\sqrt [3]{18980+35397 i \sqrt {3}} \left (666718-129789 i \sqrt {3}\right )\right ) B+\left (136488389-676743444 i \sqrt {3}+934549 \left (18980+35397 i \sqrt {3}\right )^{2/3}-\sqrt [3]{18980+35397 i \sqrt {3}} \left (26958776-6878817 i \sqrt {3}\right )\right ) C\right ) \left (x-\frac {53}{18}\right )}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right ) \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 )\right )\right )}{708588 \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 )^{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}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {1}{45} C \left (6 x^3-53 x^2+67 x+70\right )^{5/2}+\frac {\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{3/2} \left (\frac {1}{117} (9 B+53 C) \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 )^{5/2}+\frac {1}{13} \left (\frac {1}{99} \left (2106 A+\left (6201-\frac {158697}{\sqrt [3]{18980+35397 i \sqrt {3}}}-99 \sqrt [3]{18980+35397 i \sqrt {3}}\right ) B+\left (28678-\frac {934549}{\sqrt [3]{18980+35397 i \sqrt {3}}}-583 \sqrt [3]{18980+35397 i \sqrt {3}}\right ) C\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}}}} \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}+\frac {27}{11} \left (\frac {2}{7} \left (\frac {18}{5} \left (\frac {\sqrt {\frac {2}{3}} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (455 (11888793738 A+49154894001 B+245214976870 C)+\frac {\left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (85433231520 A+588536034399 B+3147821840803 C)}{\sqrt [3]{18980+35397 i \sqrt {3}}}\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 ) (85433231520 A+588536034399 B+3147821840803 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 )-\frac {1}{45} \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}}}} \left (\frac {65 \left (2569609 \left (3398602427-1343670120 i \sqrt {3}\right )+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (3398602427-1343670120 i \sqrt {3}\right )+\sqrt [3]{18980+35397 i \sqrt {3}} \left (207191147777380+94797471230919 i \sqrt {3}\right )\right ) (86562 A+303057 B+1462466 C)}{\left (18980+35397 i \sqrt {3}\right )^{8/3}}-\frac {18 \left (24570 \left (367087-3796 \sqrt [3]{18980+35397 i \sqrt {3}}+229 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) A+9 \left (3508558235-96466811 \sqrt [3]{18980+35397 i \sqrt {3}}+2188745 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) B+\left (152381644870-4765577803 \sqrt [3]{18980+35397 i \sqrt {3}}[phone] \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 ) \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 )-\frac {1}{567} \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}}}} \left (\frac {4914 \left (2519 \left (18980 i-35397 \sqrt {3}\right )+2569609 i \sqrt [3]{18980+35397 i \sqrt {3}}+\left (18980 i-35397 \sqrt {3}\right ) \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) A+9 \left (3353779 \left (18980 i-35397 \sqrt {3}\right )+1918791 \sqrt [3]{18980+35397 i \sqrt {3}} \left (1339 i+1518 \sqrt {3}\right )-1197 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (84706 i+47541 \sqrt {3}\right )\right ) B+2 \left (65837629 \left (18980 i-35397 \sqrt {3}\right )+33663 \sqrt [3]{18980+35397 i \sqrt {3}} \left (1324453 i+2292939 \sqrt {3}\right )-21 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (136214203 i+56395287 \sqrt {3}\right )\right ) C}{18980 i-35397 \sqrt {3}}-\frac {126 \left (1404 i \left (18980 i-35397 \sqrt {3}+1603 i \sqrt [3]{18980+35397 i \sqrt {3}}\right ) A+9 \left (703753-16259022 i \sqrt {3}+17633 \left (18980+35397 i \sqrt {3}\right )^{2/3}-\sqrt [3]{18980+35397 i \sqrt {3}} \left (666718-129789 i \sqrt {3}\right )\right ) B+\left (136488389-676743444 i \sqrt {3}+934549 \left (18980+35397 i \sqrt {3}\right )^{2/3}-\sqrt [3]{18980+35397 i \sqrt {3}} \left (26958776-6878817 i \sqrt {3}\right )\right ) C\right ) \left (x-\frac {53}{18}\right )}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right ) \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 )\right )\right )}{708588 \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 )^{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}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {1}{45} C \left (6 x^3-53 x^2+67 x+70\right )^{5/2}+\frac {\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{3/2} \left (\frac {1}{117} (9 B+53 C) \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 )^{5/2}+\frac {1}{13} \left (\frac {1}{99} \left (2106 A+\left (6201-\frac {158697}{\sqrt [3]{18980+35397 i \sqrt {3}}}-99 \sqrt [3]{18980+35397 i \sqrt {3}}\right ) B+\left (28678-\frac {934549}{\sqrt [3]{18980+35397 i \sqrt {3}}}-583 \sqrt [3]{18980+35397 i \sqrt {3}}\right ) C\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}}}} \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}+\frac {27}{11} \left (\frac {2}{7} \left (\frac {18}{5} \left (\frac {\left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (85433231520 A+588536034399 B+3147821840803 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 (455 (11888793738 A+49154894001 B+245214976870 C)+\frac {\left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) (85433231520 A+588536034399 B+3147821840803 C)}{\sqrt [3]{18980+35397 i \sqrt {3}}}\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 )-\frac {1}{45} \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}}}} \left (\frac {65 \left (2569609 \left (3398602427-1343670120 i \sqrt {3}\right )+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (3398602427-1343670120 i \sqrt {3}\right )+\sqrt [3]{18980+35397 i \sqrt {3}} \left (207191147777380+94797471230919 i \sqrt {3}\right )\right ) (86562 A+303057 B+1462466 C)}{\left (18980+35397 i \sqrt {3}\right )^{8/3}}-\frac {18 \left (24570 \left (367087-3796 \sqrt [3]{18980+35397 i \sqrt {3}}+229 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) A+9 \left (3508558235-96466811 \sqrt [3]{18980+35397 i \sqrt {3}}+2188745 \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) B+\left (152381644870-4765577803 \sqrt [3]{18980+35397 i \sqrt {3}}[phone] \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 ) \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 )-\frac {1}{567} \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}}}} \left (\frac {4914 \left (2519 \left (18980 i-35397 \sqrt {3}\right )+2569609 i \sqrt [3]{18980+35397 i \sqrt {3}}+\left (18980 i-35397 \sqrt {3}\right ) \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) A+9 \left (3353779 \left (18980 i-35397 \sqrt {3}\right )+1918791 \sqrt [3]{18980+35397 i \sqrt {3}} \left (1339 i+1518 \sqrt {3}\right )-1197 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (84706 i+47541 \sqrt {3}\right )\right ) B+2 \left (65837629 \left (18980 i-35397 \sqrt {3}\right )+33663 \sqrt [3]{18980+35397 i \sqrt {3}} \left (1324453 i+2292939 \sqrt {3}\right )-21 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (136214203 i+56395287 \sqrt {3}\right )\right ) C}{18980 i-35397 \sqrt {3}}-\frac {126 \left (1404 i \left (18980 i-35397 \sqrt {3}+1603 i \sqrt [3]{18980+35397 i \sqrt {3}}\right ) A+9 \left (703753-16259022 i \sqrt {3}+17633 \left (18980+35397 i \sqrt {3}\right )^{2/3}-\sqrt [3]{18980+35397 i \sqrt {3}} \left (666718-129789 i \sqrt {3}\right )\right ) B+\left (136488389-676743444 i \sqrt {3}+934549 \left (18980+35397 i \sqrt {3}\right )^{2/3}-\sqrt [3]{18980+35397 i \sqrt {3}} \left (26958776-6878817 i \sqrt {3}\right )\right ) C\right ) \left (x-\frac {53}{18}\right )}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right ) \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 )\right )\right )}{708588 \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 )^{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}}\)

Input:

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

Output:

(C*(70 + 67*x - 53*x^2 + 6*x^3)^(5/2))/45 + ((-18980 - 43281*(-53/18 + x) 
+ 2916*(-53/18 + x)^3)^(3/2)*(((9*B + 53*C)*(-((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)^(5/2))/117 + ( 
((2106*A + (6201 - 158697/(18980 + (35397*I)*Sqrt[3])^(1/3) - 99*(18980 + 
(35397*I)*Sqrt[3])^(1/3))*B + (28678 - 934549/(18980 + (35397*I)*Sqrt[3])^ 
(1/3) - 583*(18980 + (35397*I)*Sqrt[3])^(1/3))*C)*Sqrt[-((1603 + (18980 + 
(35397*I)*Sqrt[3])^(2/3))/(18980 + (35397*I)*Sqrt[3])^(1/3)) + 18*(-53/18 
+ x)]*(-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))*(-53/1 
8 + x))/(18980 + (35397*I)*Sqrt[3])^(1/3) + 324*(-53/18 + x)^2)^(5/2))/99 
+ (27*(-1/567*(Sqrt[-((1603 + (18980 + (35397*I)*Sqrt[3])^(2/3))/(18980 + 
(35397*I)*Sqrt[3])^(1/3)) + 18*(-53/18 + x)]*((4914*(2519*(18980*I - 35397 
*Sqrt[3]) + (2569609*I)*(18980 + (35397*I)*Sqrt[3])^(1/3) + (18980*I - 353 
97*Sqrt[3])*(18980 + (35397*I)*Sqrt[3])^(2/3))*A + 9*(3353779*(18980*I - 3 
5397*Sqrt[3]) + 1918791*(18980 + (35397*I)*Sqrt[3])^(1/3)*(1339*I + 1518*S 
qrt[3]) - 1197*(18980 + (35397*I)*Sqrt[3])^(2/3)*(84706*I + 47541*Sqrt[3]) 
)*B + 2*(65837629*(18980*I - 35397*Sqrt[3]) + 33663*(18980 + (35397*I)*...
 

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 1231

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) 
 - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ 
(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 
 2*p + 2))   Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* 
a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* 
c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c 
^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x 
] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] ||  !R 
ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (Integer 
Q[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 1236

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

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 = 1.12 (sec) , antiderivative size = 353, normalized size of antiderivative = 0.71

method	result	size
elliptic	\(\frac {4 C \,x^{6} \sqrt {6 x^{3}-53 x^{2}+67 x +70}}{5}+\left (\frac {12 B}{13}-\frac {1696 C}{195}\right ) x^{5} \sqrt {6 x^{3}-53 x^{2}+67 x +70}+\left (-\frac {1484 B}{143}+\frac {12 A}{11}+\frac {32423 C}{2145}\right ) x^{4} \sqrt {6 x^{3}-53 x^{2}+67 x +70}+\left (\frac {24919 B}{1287}-\frac {424 A}{33}+\frac {1314121 C}{57915}\right ) x^{3} \sqrt {6 x^{3}-53 x^{2}+67 x +70}+\left (\frac {834668 B}{27027}+\frac {2641 A}{99}+\frac {42409111 C}{2432430}\right ) x^{2} \sqrt {6 x^{3}-53 x^{2}+67 x +70}+\left (\frac {6210493 B}{270270}+\frac {4535 A}{99}+\frac {210717112 C}{1216215}\right ) x \sqrt {6 x^{3}-53 x^{2}+67 x +70}+\left (\frac {348224029 B}{1216215}+\frac {30587 A}{1782}+\frac {75793722863 C}{43783740}\right ) \sqrt {6 x^{3}-53 x^{2}+67 x +70}+\frac {\left (\frac {3986071 A}{3564}-\frac {27243620533 B}{2432430}-\frac {6140193676301 C}{87567480}\right ) \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{1311 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}+\frac {\left (\frac {1738568 A}{891}+\frac {65392892711 B}{4864860}+\frac {3147821840803 C}{43783740}\right ) \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \left (-\frac {23 \operatorname {EllipticE}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{3}+7 \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )\right )}{1311 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}\)	\(353\)
risch	\(\frac {\left (35026992 C \,x^{6}[phone] B \,x^{5}-380806272 x^{5} C [phone] x^{4} A -454371120 x^{4} B [phone] C \,x^{4}-562554720 x^{3} A [phone] B \,x^{3}[phone] C \,x^{3}[phone] A \,x^{2}[phone] B \,x^{2}[phone] C \,x^{2}[phone] A x [phone] B x [phone] C x [phone] A [phone] B [phone] C \right ) \sqrt {6 x^{3}-53 x^{2}+67 x +70}}{43783740}+\frac {\left (170866463040 A [phone] B [phone] C \right ) \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \left (-\frac {23 \operatorname {EllipticE}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{3}+7 \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )\right )}{114800966280 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}+\frac {3986071 A \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{4672404 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}-\frac {27243620533 B \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{3188915730 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}-\frac {6140193676301 C \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{114800966280 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}\)	\(368\)
default	\(A \left (\frac {12 x^{4} \sqrt {6 x^{3}-53 x^{2}+67 x +70}}{11}-\frac {424 x^{3} \sqrt {6 x^{3}-53 x^{2}+67 x +70}}{33}+\frac {2641 x^{2} \sqrt {6 x^{3}-53 x^{2}+67 x +70}}{99}+\frac {4535 x \sqrt {6 x^{3}-53 x^{2}+67 x +70}}{99}+\frac {30587 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}{1782}+\frac {3986071 \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 )}{4672404 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}+\frac {1738568 \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 )}{1168101 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}\right )+B \left (\frac {12 x^{5} \sqrt {6 x^{3}-53 x^{2}+67 x +70}}{13}-\frac {1484 x^{4} \sqrt {6 x^{3}-53 x^{2}+67 x +70}}{143}+\frac {24919 x^{3} \sqrt {6 x^{3}-53 x^{2}+67 x +70}}{1287}+\frac {834668 x^{2} \sqrt {6 x^{3}-53 x^{2}+67 x +70}}{27027}+\frac {6210493 x \sqrt {6 x^{3}-53 x^{2}+67 x +70}}{270270}+\frac {348224029 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}{1216215}-\frac {27243620533 \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 )}{3188915730 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}+\frac {65392892711 \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 )}{6377831460 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}\right )+C \left (\frac {4 x^{6} \sqrt {6 x^{3}-53 x^{2}+67 x +70}}{5}-\frac {1696 x^{5} \sqrt {6 x^{3}-53 x^{2}+67 x +70}}{195}+\frac {32423 x^{4} \sqrt {6 x^{3}-53 x^{2}+67 x +70}}{2145}+\frac {1314121 x^{3} \sqrt {6 x^{3}-53 x^{2}+67 x +70}}{57915}+\frac {42409111 x^{2} \sqrt {6 x^{3}-53 x^{2}+67 x +70}}{2432430}+\frac {210717112 x \sqrt {6 x^{3}-53 x^{2}+67 x +70}}{1216215}+\frac {75793722863 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}{43783740}-\frac {6140193676301 \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 )}{114800966280 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}+\frac {3147821840803 \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 )}{57400483140 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}\right )\)	\(782\)

Input:

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

Output:

4/5*C*x^6*(6*x^3-53*x^2+67*x+70)^(1/2)+(12/13*B-1696/195*C)*x^5*(6*x^3-53* 
x^2+67*x+70)^(1/2)+(-1484/143*B+12/11*A+32423/2145*C)*x^4*(6*x^3-53*x^2+67 
*x+70)^(1/2)+(24919/1287*B-424/33*A+1314121/57915*C)*x^3*(6*x^3-53*x^2+67* 
x+70)^(1/2)+(834668/27027*B+2641/99*A[phone]/2432430*C)*x^2*(6*x^3-53*x^ 
2+67*x+70)^(1/2)+(6210493/270270*B+4535/99*A[phone]/1216215*C)*x*(6*x^3 
-53*x^2+67*x+70)^(1/2)+(348224029/1216215*B+30587/1782*A[phone]/43783 
740*C)*(6*x^3-53*x^2+67*x+70)^(1/2)+1/1311*(3986071/3564*A-27243620533/243 
2430*B-6140193676301/87567480*C)*(76+114*x)^(1/2)*(483-69*x)^(1/2)*(285-11 
4*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*(1738568/891*A[phone]/4864860*B[phone]/43 
783740*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*874^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.30 \[ \int \left (A+B x+C x^2\right ) \left (70+67 x-53 x^2+6 x^3\right )^{3/2} \, dx=\frac {1}{5196312} \, \sqrt {6} {\left (11888793738 \, A + 49154894001 \, B + 245214976870 \, C\right )} {\rm weierstrassPInverse}\left (\frac {1603}{27}, \frac {18980}{729}, x - \frac {53}{18}\right ) - \frac {1}{131351220} \, \sqrt {6} {\left (85433231520 \, A + 588536034399 \, B + 3147821840803 \, C\right )} {\rm weierstrassZeta}\left (\frac {1603}{27}, \frac {18980}{729}, {\rm weierstrassPInverse}\left (\frac {1603}{27}, \frac {18980}{729}, x - \frac {53}{18}\right )\right ) + \frac {1}{43783740} \, {\left (35026992 \, C x^{6} + 898128 \, {\left (45 \, B - 424 \, C\right )} x^{5} + 20412 \, {\left (2340 \, A - 22260 \, B + 32423 \, C\right )} x^{4} - 756 \, {\left (744120 \, A - 1121355 \, B - 1314121 \, C\right )} x^{3} + 18 \, {\left (64889370 \, A + 75120120 \, B + 42409111 \, C\right )} x^{2} + 18 \, {\left (111424950 \, A + 55894437 \, B + 421434224 \, C\right )} x + 751522590 \, A + 12536065044 \, B + 75793722863 \, C\right )} \sqrt {6 \, x^{3} - 53 \, x^{2} + 67 \, x + 70} \] Input:

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

Output:

1/5196312*sqrt(6)*(11888793738*A + 49154894001*B + 245214976870*C)*weierst 
rassPInverse(1603/27, 18980/729, x - 53/18) - 1/131351220*sqrt(6)*(8543323 
1520*A + 588536034399*B + 3147821840803*C)*weierstrassZeta(1603/27, 18980/ 
729, weierstrassPInverse(1603/27, 18980/729, x - 53/18)) + 1/43783740*(350 
26992*C*x^6 + 898128*(45*B - 424*C)*x^5 + 20412*(2340*A - 22260*B + 32423* 
C)*x^4 - 756*(744120*A - 1121355*B - 1314121*C)*x^3 + 18*(64889370*A + 751 
20120*B + 42409111*C)*x^2 + 18*(111424950*A + 55894437*B + 421434224*C)*x 
+ 751522590*A + 12536065044*B + 75793722863*C)*sqrt(6*x^3 - 53*x^2 + 67*x 
+ 70)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 13.53 (sec) , antiderivative size = 913, normalized size of antiderivative = 1.83 \[ \int \left (A+B x+C x^2\right ) \left (70+67 x-53 x^2+6 x^3\right )^{3/2} \, dx=\text {Too large to display} \] Input:

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

Output:

(1070545*A)/(891*(67*x - 53*x^2 + 6*x^3 + 70)^(1/2)) + (696448058*B)/(3474 
9*(67*x - 53*x^2 + 6*x^3 + 70)^(1/2)) + (75793722863*C)/(625482*(67*x - 53 
*x^2 + 6*x^3 + 70)^(1/2)) + (7763429*A*x)/(1782*(67*x - 53*x^2 + 6*x^3 + 7 
0)^(1/2)) + (25287315238*B*x)/(1216215*(67*x - 53*x^2 + 6*x^3 + 70)^(1/2)) 
 + (5609186554061*C*x)/(43783740*(67*x - 53*x^2 + 6*x^3 + 70)^(1/2)) + (71 
75759*A*x^2)/(1782*(67*x - 53*x^2 + 6*x^3 + 70)^(1/2)) - (426757*A*x^3)/(2 
97*(67*x - 53*x^2 + 6*x^3 + 70)^(1/2)) - (190427*A*x^4)/(99*(67*x - 53*x^2 
 + 6*x^3 + 70)^(1/2)) + (30166*A*x^5)/(33*(67*x - 53*x^2 + 6*x^3 + 70)^(1/ 
2)) - (1484*A*x^6)/(11*(67*x - 53*x^2 + 6*x^3 + 70)^(1/2)) + (72*A*x^7)/(1 
1*(67*x - 53*x^2 + 6*x^3 + 70)^(1/2)) - (5581682279*B*x^2)/(486486*(67*x - 
 53*x^2 + 6*x^3 + 70)^(1/2)) + (3182038309*B*x^3)/(810810*(67*x - 53*x^2 + 
 6*x^3 + 70)^(1/2)) - (125416976*B*x^4)/(135135*(67*x - 53*x^2 + 6*x^3 + 7 
0)^(1/2)) - (13257457*B*x^5)/(9009*(67*x - 53*x^2 + 6*x^3 + 70)^(1/2)) + ( 
312326*B*x^6)/(429*(67*x - 53*x^2 + 6*x^3 + 70)^(1/2)) - (15900*B*x^7)/(14 
3*(67*x - 53*x^2 + 6*x^3 + 70)^(1/2)) + (72*B*x^8)/(13*(67*x - 53*x^2 + 6* 
x^3 + 70)^(1/2)) - (691076431547*C*x^2)/(8756748*(67*x - 53*x^2 + 6*x^3 + 
70)^(1/2)) + (14450229889*C*x^3)/(3648645*(67*x - 53*x^2 + 6*x^3 + 70)^(1/ 
2)) + (1310519339*C*x^4)/(486486*(67*x - 53*x^2 + 6*x^3 + 70)^(1/2)) - (28 
1376211*C*x^5)/(405405*(67*x - 53*x^2 + 6*x^3 + 70)^(1/2)) - (23006017*C*x 
^6)/(19305*(67*x - 53*x^2 + 6*x^3 + 70)^(1/2)) + (1298278*C*x^7)/(2145*...
 

Reduce [F]

\[ \int \left (A+B x+C x^2\right ) \left (70+67 x-53 x^2+6 x^3\right )^{3/2} \, dx=\frac {12 \sqrt {6 x^{3}-53 x^{2}+67 x +70}\, a \,x^{4}}{11}-\frac {424 \sqrt {6 x^{3}-53 x^{2}+67 x +70}\, a \,x^{3}}{33}+\frac {2641 \sqrt {6 x^{3}-53 x^{2}+67 x +70}\, a \,x^{2}}{99}+\frac {4535 \sqrt {6 x^{3}-53 x^{2}+67 x +70}\, a x}{99}-\frac {206225 \sqrt {6 x^{3}-53 x^{2}+67 x +70}\, a}{10494}+\frac {12 \sqrt {6 x^{3}-53 x^{2}+67 x +70}\, b \,x^{5}}{13}-\frac {1484 \sqrt {6 x^{3}-53 x^{2}+67 x +70}\, b \,x^{4}}{143}+\frac {24919 \sqrt {6 x^{3}-53 x^{2}+67 x +70}\, b \,x^{3}}{1287}+\frac {834668 \sqrt {6 x^{3}-53 x^{2}+67 x +70}\, b \,x^{2}}{27027}+\frac {6210493 \sqrt {6 x^{3}-53 x^{2}+67 x +70}\, b x}{270270}+\frac {936733493 \sqrt {6 x^{3}-53 x^{2}+67 x +70}\, b}{28648620}+\frac {4 \sqrt {6 x^{3}-53 x^{2}+67 x +70}\, c \,x^{6}}{5}-\frac {1696 \sqrt {6 x^{3}-53 x^{2}+67 x +70}\, c \,x^{5}}{195}+\frac {32423 \sqrt {6 x^{3}-53 x^{2}+67 x +70}\, c \,x^{4}}{2145}+\frac {1314121 \sqrt {6 x^{3}-53 x^{2}+67 x +70}\, c \,x^{3}}{57915}+\frac {42409111 \sqrt {6 x^{3}-53 x^{2}+67 x +70}\, c \,x^{2}}{2432430}+\frac {210717112 \sqrt {6 x^{3}-53 x^{2}+67 x +70}\, c x}{1216215}+\frac {24145707526 \sqrt {6 x^{3}-53 x^{2}+67 x +70}\, c}{64459395}+\frac {49358875 \left (\int \frac {\sqrt {6 x^{3}-53 x^{2}+67 x +70}}{6 x^{3}-53 x^{2}+67 x +70}d x \right ) a}{20988}-\frac {154924860151 \left (\int \frac {\sqrt {6 x^{3}-53 x^{2}+67 x +70}}{6 x^{3}-53 x^{2}+67 x +70}d x \right ) b}{57297240}-\frac {1590641687641 \left (\int \frac {\sqrt {6 x^{3}-53 x^{2}+67 x +70}}{6 x^{3}-53 x^{2}+67 x +70}d x \right ) c}{64459395}+\frac {1738568 \left (\int \frac {\sqrt {6 x^{3}-53 x^{2}+67 x +70}\, x^{2}}{6 x^{3}-53 x^{2}+67 x +70}d x \right ) a}{5247}+\frac {65392892711 \left (\int \frac {\sqrt {6 x^{3}-53 x^{2}+67 x +70}\, x^{2}}{6 x^{3}-53 x^{2}+67 x +70}d x \right ) b}{28648620}+\frac {3147821840803 \left (\int \frac {\sqrt {6 x^{3}-53 x^{2}+67 x +70}\, x^{2}}{6 x^{3}-53 x^{2}+67 x +70}d x \right ) c}{257837580} \] Input:

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

Output:

(562554720*sqrt(6*x**3 - 53*x**2 + 67*x + 70)*a*x**4 - 6625644480*sqrt(6*x 
**3 - 53*x**2 + 67*x + 70)*a*x**3 + 13756546440*sqrt(6*x**3 - 53*x**2 + 67 
*x + 70)*a*x**2 + 23622089400*sqrt(6*x**3 - 53*x**2 + 67*x + 70)*a*x - 101 
33896500*sqrt(6*x**3 - 53*x**2 + 67*x + 70)*a + 476007840*sqrt(6*x**3 - 53 
*x**2 + 67*x + 70)*b*x**5 - 5351482080*sqrt(6*x**3 - 53*x**2 + 67*x + 70)* 
b*x**4 + 9984544920*sqrt(6*x**3 - 53*x**2 + 67*x + 70)*b*x**3 + 1592546544 
0*sqrt(6*x**3 - 53*x**2 + 67*x + 70)*b*x**2 + 11849620644*sqrt(6*x**3 - 53 
*x**2 + 67*x + 70)*b*x + 16861202874*sqrt(6*x**3 - 53*x**2 + 67*x + 70)*b 
+ 412540128*sqrt(6*x**3 - 53*x**2 + 67*x + 70)*c*x**6 - 4485051648*sqrt(6* 
x**3 - 53*x**2 + 67*x + 70)*c*x**5 + 7794748584*sqrt(6*x**3 - 53*x**2 + 67 
*x + 70)*c*x**4 + 11700933384*sqrt(6*x**3 - 53*x**2 + 67*x + 70)*c*x**3 + 
8990731532*sqrt(6*x**3 - 53*x**2 + 67*x + 70)*c*x**2 + 89344055488*sqrt(6* 
x**3 - 53*x**2 + 67*x + 70)*c*x + 193165660208*sqrt(6*x**3 - 53*x**2 + 67* 
x + 70)*c + 1212747558750*int(sqrt(6*x**3 - 53*x**2 + 67*x + 70)/(6*x**3 - 
 53*x**2 + 67*x + 70),x)*a - 1394323741359*int(sqrt(6*x**3 - 53*x**2 + 67* 
x + 70)/(6*x**3 - 53*x**2 + 67*x + 70),x)*b - 12725133501128*int(sqrt(6*x* 
*3 - 53*x**2 + 67*x + 70)/(6*x**3 - 53*x**2 + 67*x + 70),x)*c + 1708664630 
40*int((sqrt(6*x**3 - 53*x**2 + 67*x + 70)*x**2)/(6*x**3 - 53*x**2 + 67*x 
+ 70),x)*a + 1177072068798*int((sqrt(6*x**3 - 53*x**2 + 67*x + 70)*x**2)/( 
6*x**3 - 53*x**2 + 67*x + 70),x)*b + 6295643681606*int((sqrt(6*x**3 - 5...