\(\int \frac {1}{(70+67 x-53 x^2+6 x^3)^{5/2}} \, dx\) [134]

Optimal result

Integrand size = 19, antiderivative size = 397 \[ \int \frac {1}{\left (70+67 x-53 x^2+6 x^3\right )^{5/2}} \, dx=-\frac {2 (5-2 x) (7-x) (2+3 x)}{621 \left (70+67 x-53 x^2+6 x^3\right )^{5/2}}-\frac {292 (5-2 x) (7-x)^2 (2+3 x)}{42849 \left (70+67 x-53 x^2+6 x^3\right )^{5/2}}+\frac {45248 (5-2 x) (7-x)^3 (2+3 x)}{21981537 \left (70+67 x-53 x^2+6 x^3\right )^{5/2}}-\frac {680680 (5-2 x)^2 (7-x)^3 (2+3 x)}{3758842827 \left (70+67 x-53 x^2+6 x^3\right )^{5/2}}-\frac {624820 (5-2 x)^3 (7-x)^3 (2+3 x)}{547538105133 \left (70+67 x-53 x^2+6 x^3\right )^{5/2}}-\frac {973598080 (5-2 x)^3 (7-x)^3 (2+3 x)^2}{239274151943121 \left (70+67 x-53 x^2+6 x^3\right )^{5/2}}-\frac {973598080 (5-2 x)^{5/2} (7-x)^{5/2} (2+3 x)^{5/2} E\left (\arcsin \left (\frac {\sqrt {2+3 x}}{\sqrt {23}}\right )|\frac {46}{19}\right )}{37780129254177 \sqrt {19} \left (70+67 x-53 x^2+6 x^3\right )^{5/2}}-\frac {34994680 (-7+x)^{5/2} (-5+2 x)^{5/2} (2+3 x)^{5/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {19}{2}}}{\sqrt {2+3 x}}\right ),\frac {46}{19}\right )}{1399264046451 \sqrt {19} \left (70+67 x-53 x^2+6 x^3\right )^{5/2}} \] Output:

-2/621*(5-2*x)*(7-x)*(2+3*x)/(6*x^3-53*x^2+67*x+70)^(5/2)-292/42849*(5-2*x 
)*(7-x)^2*(2+3*x)/(6*x^3-53*x^2+67*x+70)^(5/2)+45248/21981537*(5-2*x)*(7-x 
)^3*(2+3*x)/(6*x^3-53*x^2+67*x+70)^(5/2)-680680/3758842827*(5-2*x)^2*(7-x) 
^3*(2+3*x)/(6*x^3-53*x^2+67*x+70)^(5/2)-624820/547538105133*(5-2*x)^3*(7-x 
)^3*(2+3*x)/(6*x^3-53*x^2+67*x+70)^(5/2)-973598080/239274151943121*(5-2*x) 
^3*(7-x)^3*(2+3*x)^2/(6*x^3-53*x^2+67*x+70)^(5/2)-973598080/71782245582936 
3*(5-2*x)^(5/2)*(7-x)^(5/2)*(2+3*x)^(5/2)*EllipticE(1/23*(2+3*x)^(1/2)*23^ 
(1/2),1/19*874^(1/2))*19^(1/2)/(6*x^3-53*x^2+67*x+70)^(5/2)-34994680/26586 
016882569*(-7+x)^(5/2)*(-5+2*x)^(5/2)*(2+3*x)^(5/2)*EllipticF(1/2*38^(1/2) 
/(2+3*x)^(1/2),1/19*874^(1/2))*19^(1/2)/(6*x^3-53*x^2+67*x+70)^(5/2)
 

Mathematica [A] (verified)

Time = 9.09 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.41 \[ \int \frac {1}{\left (70+67 x-53 x^2+6 x^3\right )^{5/2}} \, dx=-\frac {2 \left (15468489 \left (13687-125932 x+19236 x^2\right )+70 \left (381044737-392492842 x+41725632 x^2\right ) \left (70+67 x-53 x^2+6 x^3\right )+70 \sqrt {46} \sqrt {5-2 x} \sqrt {7-x} \sqrt {2+3 x} \left (70+67 x-53 x^2+6 x^3\right ) \left (6954272 E\left (\arcsin \left (\sqrt {\frac {2}{19}} \sqrt {2+3 x}\right )|\frac {19}{46}\right )-6869475 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{19}} \sqrt {2+3 x}\right ),\frac {19}{46}\right )\right )\right )}{717822455829363 \left (70+67 x-53 x^2+6 x^3\right )^{3/2}} \] Input:

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

Output:

(-2*(15468489*(13687 - 125932*x + 19236*x^2) + 70*(381044737 - 392492842*x 
 + 41725632*x^2)*(70 + 67*x - 53*x^2 + 6*x^3) + 70*Sqrt[46]*Sqrt[5 - 2*x]* 
Sqrt[7 - x]*Sqrt[2 + 3*x]*(70 + 67*x - 53*x^2 + 6*x^3)*(6954272*EllipticE[ 
ArcSin[Sqrt[2/19]*Sqrt[2 + 3*x]], 19/46] - 6869475*EllipticF[ArcSin[Sqrt[2 
/19]*Sqrt[2 + 3*x]], 19/46])))/(717822455829363*(70 + 67*x - 53*x^2 + 6*x^ 
3)^(3/2))
 

Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 4.41 (sec) , antiderivative size = 2489, normalized size of antiderivative = 6.27, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.789, Rules used = {2481, 2475, 27, 1165, 27, 1235, 27, 1237, 27, 1237, 27, 1269, 1172, 321, 327}

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

\(\displaystyle \int \frac {1}{\left (6 x^3-53 x^2+67 x+70\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 2481

\(\displaystyle \int \frac {1}{\left (6 \left (x-\frac {53}{18}\right )^3-\frac {1603}{18} \left (x-\frac {53}{18}\right )-\frac {9490}{243}\right )^{5/2}}d\left (x-\frac {53}{18}\right )\)

\(\Big \downarrow \) 2475

\(\displaystyle \frac {972 \sqrt {2} \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{5/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{5/2} \int \frac {2187 \sqrt {3}}{\left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{5/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{5/2}}d\left (x-\frac {53}{18}\right )}{\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2125764 \sqrt {6} \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{5/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{5/2} \int \frac {1}{\left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{5/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{5/2}}d\left (x-\frac {53}{18}\right )}{\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{5/2}}\)

\(\Big \downarrow \) 1165

\(\displaystyle \frac {2125764 \sqrt {6} \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{5/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{5/2} \left (\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \int -\frac {1102248 \left (\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}}+916\right )}{\left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{5/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{3/2}}d\left (x-\frac {53}{18}\right )}{100327557744 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}-\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (1603+\frac {9 \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}}}\right )}{2867157 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{3/2}}\right )}{\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2125764 \sqrt {6} \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{5/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{5/2} \left (-\frac {7 i \sqrt [3]{18980+35397 i \sqrt {3}} \int \frac {\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}}+916}{\left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{5/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{3/2}}d\left (x-\frac {53}{18}\right )}{637146 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}-\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (1603+\frac {9 \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}}}\right )}{2867157 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{3/2}}\right )}{\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{5/2}}\)

\(\Big \downarrow \) 1235

\(\displaystyle \frac {2125764 \sqrt {6} \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{5/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{5/2} \left (-\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (\frac {9 \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}}}+1603\right )}{2867157 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{3/2}}-\frac {7 i \sqrt [3]{18980+35397 i \sqrt {3}} \left (\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \int -\frac {1889568 \left (5 \left (367087-\frac {\left (624464501 i-67183506 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}}{18980 i-35397 \sqrt {3}}+\frac {229 \left (3146434147 i+1813954662 \sqrt {3}\right )}{\left (18980 i-35397 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}}\right )-\frac {9 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (2569609-458 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) \left (x-\frac {53}{18}\right )}{18980+35397 i \sqrt {3}}\right )}{\left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{5/2} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}d\left (x-\frac {53}{18}\right )}{33442519248 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}-\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (-\frac {18 \left (2569609 i+\left (18980 i-35397 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}-458 i \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{18980 i-35397 \sqrt {3}}+\frac {1603 \left (3038362027 i+2015505180 \sqrt {3}\right )}{\left (18980 i-35397 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}}-\frac {\left (4479323627 i-671835060 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}}{18980 i-35397 \sqrt {3}}+1468348\right )}{955719 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}\right )}{637146 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}\right )}{\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2125764 \sqrt {6} \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{5/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{5/2} \left (-\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (\frac {9 \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}}}+1603\right )}{2867157 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{3/2}}-\frac {7 i \sqrt [3]{18980+35397 i \sqrt {3}} \left (-\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (-\frac {18 \left (2569609 i+\left (18980 i-35397 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}-458 i \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{18980 i-35397 \sqrt {3}}+\frac {1603 \left (3038362027 i+2015505180 \sqrt {3}\right )}{\left (18980 i-35397 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}}-\frac {\left (4479323627 i-671835060 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}}{18980 i-35397 \sqrt {3}}+1468348\right )}{955719 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}-\frac {2 i \sqrt [3]{18980+35397 i \sqrt {3}} \int \frac {5 \left (367087-\frac {\left (624464501 i-67183506 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}}{18980 i-35397 \sqrt {3}}+\frac {229 \left (3146434147 i+1813954662 \sqrt {3}\right )}{\left (18980 i-35397 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}}\right )-\frac {9 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (2569609-458 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) \left (x-\frac {53}{18}\right )}{18980+35397 i \sqrt {3}}}{\left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{5/2} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}d\left (x-\frac {53}{18}\right )}{35397 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}\right )}{637146 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}\right )}{\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{5/2}}\)

\(\Big \downarrow \) 1237

\(\displaystyle \frac {2125764 \sqrt {6} \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{5/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{5/2} \left (-\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (\frac {9 \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}}}+1603\right )}{2867157 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{3/2}}-\frac {7 i \sqrt [3]{18980+35397 i \sqrt {3}} \left (-\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (-\frac {18 \left (2569609 i+\left (18980 i-35397 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}-458 i \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{18980 i-35397 \sqrt {3}}+\frac {1603 \left (3038362027 i+2015505180 \sqrt {3}\right )}{\left (18980 i-35397 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}}-\frac {\left (4479323627 i-671835060 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}}{18980 i-35397 \sqrt {3}}+1468348\right )}{955719 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}-\frac {2 i \sqrt [3]{18980+35397 i \sqrt {3}} \left (\frac {5 \left (\left (18980+35397 i \sqrt {3}\right )^{2/3} \left (220162847+44789004 i \sqrt {3}\right )-229 \left (964755289-761413068 i \sqrt {3}\right )\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}}{18 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2}}-\frac {\left (18980+35397 i \sqrt {3}\right )^{2/3} \int -\frac {729 i \left (\frac {\left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (4405044 \left (18980 i-35397 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}-\left (18980+35397 i \sqrt {3}\right )^{2/3} \left (5960471297 i+223945020 \sqrt {3}\right )+229 \left (35959452679 i+12316976100 \sqrt {3}\right )\right )}{\left (18980+35397 i \sqrt {3}\right )^{5/3}}-\frac {90 \left (\left (220162847 i-44789004 \sqrt {3}\right ) \left (18980+35397 i \sqrt {3}\right )^{2/3}-229 \left (964755289 i+761413068 \sqrt {3}\right )\right ) \left (x-\frac {53}{18}\right )}{\left (18980+35397 i \sqrt {3}\right )^{4/3}}\right )}{\left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}d\left (x-\frac {53}{18}\right )}{1458 \left (2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right )}\right )}{35397 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}\right )}{637146 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}\right )}{\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2125764 \sqrt {6} \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{5/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{5/2} \left (-\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (\frac {9 \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}}}+1603\right )}{2867157 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{3/2}}-\frac {7 i \sqrt [3]{18980+35397 i \sqrt {3}} \left (-\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (-\frac {18 \left (2569609 i+\left (18980 i-35397 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}-458 i \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{18980 i-35397 \sqrt {3}}+\frac {1603 \left (3038362027 i+2015505180 \sqrt {3}\right )}{\left (18980 i-35397 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}}-\frac {\left (4479323627 i-671835060 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}}{18980 i-35397 \sqrt {3}}+1468348\right )}{955719 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}-\frac {2 i \sqrt [3]{18980+35397 i \sqrt {3}} \left (\frac {i \left (18980+35397 i \sqrt {3}\right )^{2/3} \int \frac {\frac {\left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (4405044 \left (18980 i-35397 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}-\left (18980+35397 i \sqrt {3}\right )^{2/3} \left (5960471297 i+223945020 \sqrt {3}\right )+229 \left (35959452679 i+12316976100 \sqrt {3}\right )\right )}{\left (18980+35397 i \sqrt {3}\right )^{5/3}}-\frac {90 \left (\left (220162847 i-44789004 \sqrt {3}\right ) \left (18980+35397 i \sqrt {3}\right )^{2/3}-229 \left (964755289 i+761413068 \sqrt {3}\right )\right ) \left (x-\frac {53}{18}\right )}{\left (18980+35397 i \sqrt {3}\right )^{4/3}}}{\left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}d\left (x-\frac {53}{18}\right )}{2 \left (2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right )}+\frac {5 \left (\left (18980+35397 i \sqrt {3}\right )^{2/3} \left (220162847+44789004 i \sqrt {3}\right )-229 \left (964755289-761413068 i \sqrt {3}\right )\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}}{18 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2}}\right )}{35397 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}\right )}{637146 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}\right )}{\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{5/2}}\)

\(\Big \downarrow \) 1237

\(\displaystyle \frac {2125764 \sqrt {6} \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{5/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{5/2} \left (-\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (\frac {9 \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}}}+1603\right )}{2867157 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{3/2}}-\frac {7 i \sqrt [3]{18980+35397 i \sqrt {3}} \left (-\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (-\frac {18 \left (2569609 i+\left (18980 i-35397 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}-458 i \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{18980 i-35397 \sqrt {3}}+\frac {1603 \left (3038362027 i+2015505180 \sqrt {3}\right )}{\left (18980 i-35397 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}}-\frac {\left (4479323627 i-671835060 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}}{18980 i-35397 \sqrt {3}}+1468348\right )}{955719 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}-\frac {2 i \sqrt [3]{18980+35397 i \sqrt {3}} \left (\frac {i \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (\frac {34771360 i \left (18980+35397 i \sqrt {3}\right )^{2/3} \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}}{9 \left (2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}}}-\frac {\left (18980+35397 i \sqrt {3}\right )^{2/3} \int \frac {4860 i \left (62588448 \left (x-\frac {53}{18}\right )[phone]\right )}{\sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}d\left (x-\frac {53}{18}\right )}{486 \left (2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right )}\right )}{2 \left (2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right )}+\frac {5 \left (\left (18980+35397 i \sqrt {3}\right )^{2/3} \left (220162847+44789004 i \sqrt {3}\right )-229 \left (964755289-761413068 i \sqrt {3}\right )\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}}{18 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2}}\right )}{35397 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}\right )}{637146 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}\right )}{\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2125764 \sqrt {6} \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{5/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{5/2} \left (-\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (\frac {9 \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}}}+1603\right )}{2867157 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{3/2}}-\frac {7 i \sqrt [3]{18980+35397 i \sqrt {3}} \left (-\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (-\frac {18 \left (2569609 i+\left (18980 i-35397 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}-458 i \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{18980 i-35397 \sqrt {3}}+\frac {1603 \left (3038362027 i+2015505180 \sqrt {3}\right )}{\left (18980 i-35397 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}}-\frac {\left (4479323627 i-671835060 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}}{18980 i-35397 \sqrt {3}}+1468348\right )}{955719 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}-\frac {2 i \sqrt [3]{18980+35397 i \sqrt {3}} \left (\frac {i \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (\frac {34771360 i \left (18980+35397 i \sqrt {3}\right )^{2/3} \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}}{9 \left (2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}}}-\frac {10 i \left (18980+35397 i \sqrt {3}\right )^{2/3} \int \frac {62588448 \left (x-\frac {53}{18}\right )[phone]}{\sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}d\left (x-\frac {53}{18}\right )}{2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}}\right )}{2 \left (2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right )}+\frac {5 \left (\left (18980+35397 i \sqrt {3}\right )^{2/3} \left (220162847+44789004 i \sqrt {3}\right )-229 \left (964755289-761413068 i \sqrt {3}\right )\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}}{18 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2}}\right )}{35397 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}\right )}{637146 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}\right )}{\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{5/2}}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {2125764 \sqrt {6} \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{5/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{5/2} \left (-\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (\frac {9 \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}}}+1603\right )}{2867157 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{3/2}}-\frac {7 i \sqrt [3]{18980+35397 i \sqrt {3}} \left (-\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (-\frac {18 \left (2569609 i+\left (18980 i-35397 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}-458 i \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{18980 i-35397 \sqrt {3}}+\frac {1603 \left (3038362027 i+2015505180 \sqrt {3}\right )}{\left (18980 i-35397 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}}-\frac {\left (4479323627 i-671835060 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}}{18980 i-35397 \sqrt {3}}+1468348\right )}{955719 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}-\frac {2 i \sqrt [3]{18980+35397 i \sqrt {3}} \left (\frac {i \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (\frac {34771360 i \left (18980+35397 i \sqrt {3}\right )^{2/3} \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}}{9 \left (2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}}}-\frac {10 i \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (\left (220162847+\frac {3477136 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}{\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 )+3477136 \int \frac {\sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}}}{\sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}d\left (x-\frac {53}{18}\right )\right )}{2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}}\right )}{2 \left (2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right )}+\frac {5 \left (\left (18980+35397 i \sqrt {3}\right )^{2/3} \left (220162847+44789004 i \sqrt {3}\right )-229 \left (964755289-761413068 i \sqrt {3}\right )\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}}{18 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2}}\right )}{35397 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}\right )}{637146 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}\right )}{\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{5/2}}\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {2125764 \sqrt {6} \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{5/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{5/2} \left (-\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (\frac {9 \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}}}+1603\right )}{2867157 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{3/2}}-\frac {7 i \sqrt [3]{18980+35397 i \sqrt {3}} \left (-\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (-\frac {18 \left (2569609 i+\left (18980 i-35397 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}-458 i \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{18980 i-35397 \sqrt {3}}+\frac {1603 \left (3038362027 i+2015505180 \sqrt {3}\right )}{\left (18980 i-35397 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}}-\frac {\left (4479323627 i-671835060 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}}{18980 i-35397 \sqrt {3}}+1468348\right )}{955719 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}-\frac {2 i \sqrt [3]{18980+35397 i \sqrt {3}} \left (\frac {i \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (\frac {34771360 i \left (18980+35397 i \sqrt {3}\right )^{2/3} \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}}{9 \left (2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}}}-\frac {10 i \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (-\frac {\sqrt {\frac {2}{3}} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (220162847+\frac {3477136 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}{\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 {1738568 \sqrt {\frac {2}{3}} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \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 {-\left (18980+35397 i \sqrt {3}\right )^{2/3}} \sqrt {-\frac {\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}-18 \left (x-\frac {53}{18}\right )}{\frac {1}{324} \sqrt {-\frac {1}{3} \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (324-\frac {519372}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}\right )}{2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}}\right )}{2 \left (2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right )}+\frac {5 \left (\left (18980+35397 i \sqrt {3}\right )^{2/3} \left (220162847+44789004 i \sqrt {3}\right )-229 \left (964755289-761413068 i \sqrt {3}\right )\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}}{18 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2}}\right )}{35397 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}\right )}{637146 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}\right )}{\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{5/2}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {2125764 \sqrt {6} \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{5/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{5/2} \left (-\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (\frac {9 \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}}}+1603\right )}{2867157 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{3/2}}-\frac {7 i \sqrt [3]{18980+35397 i \sqrt {3}} \left (-\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (-\frac {18 \left (2569609 i+\left (18980 i-35397 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}-458 i \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{18980 i-35397 \sqrt {3}}+\frac {1603 \left (3038362027 i+2015505180 \sqrt {3}\right )}{\left (18980 i-35397 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}}-\frac {\left (4479323627 i-671835060 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}}{18980 i-35397 \sqrt {3}}+1468348\right )}{955719 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}-\frac {2 i \sqrt [3]{18980+35397 i \sqrt {3}} \left (\frac {i \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (\frac {34771360 i \left (18980+35397 i \sqrt {3}\right )^{2/3} \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}}{9 \left (2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}}}-\frac {10 i \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (\frac {\sqrt {\frac {2}{3}} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (220162847+\frac {3477136 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}{\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 {1738568 \sqrt {\frac {2}{3}} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \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 {-\left (18980+35397 i \sqrt {3}\right )^{2/3}} \sqrt {-\frac {\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}-18 \left (x-\frac {53}{18}\right )}{\frac {1}{324} \sqrt {-\frac {1}{3} \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (324-\frac {519372}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}\right )}{2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}}\right )}{2 \left (2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right )}+\frac {5 \left (\left (18980+35397 i \sqrt {3}\right )^{2/3} \left (220162847+44789004 i \sqrt {3}\right )-229 \left (964755289-761413068 i \sqrt {3}\right )\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}}{18 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2}}\right )}{35397 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}\right )}{637146 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}\right )}{\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{5/2}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {2125764 \sqrt {6} \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{5/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{5/2} \left (-\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (\frac {9 \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}}}+1603\right )}{2867157 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \left (324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603\right )^{3/2}}-\frac {7 i \sqrt [3]{18980+35397 i \sqrt {3}} \left (-\frac {i \sqrt [3]{18980+35397 i \sqrt {3}} \left (-\frac {18 \left (2569609 i+\left (18980 i-35397 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}-458 i \left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{18980 i-35397 \sqrt {3}}+\frac {1603 \left (3038362027 i+2015505180 \sqrt {3}\right )}{\left (18980 i-35397 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}}-\frac {\left (4479323627 i-671835060 \sqrt {3}\right ) \sqrt [3]{18980+35397 i \sqrt {3}}}{18980 i-35397 \sqrt {3}}+1468348\right )}{955719 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}-\frac {2 i \sqrt [3]{18980+35397 i \sqrt {3}} \left (\frac {i \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (\frac {34771360 i \left (18980+35397 i \sqrt {3}\right )^{2/3} \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}}{9 \left (2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}}}-\frac {10 i \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (\frac {1738568 \sqrt {\frac {2}{3}} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \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 {-\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 (220162847+\frac {3477136 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}{\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 )}{2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}}\right )}{2 \left (2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right )}+\frac {5 \left (\left (18980+35397 i \sqrt {3}\right )^{2/3} \left (220162847+44789004 i \sqrt {3}\right )-229 \left (964755289-761413068 i \sqrt {3}\right )\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}}{18 \left (18980+35397 i \sqrt {3}\right )^{2/3} \left (2569609+1603 \left (18980+35397 i \sqrt {3}\right )^{2/3}+\left (18980+35397 i \sqrt {3}\right )^{4/3}\right ) \left (18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}\right )^{3/2}}\right )}{35397 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}\right )}{637146 \sqrt {3} \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}\right )}{\left (2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980\right )^{5/2}}\)

Input:

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

Output:

(2125764*Sqrt[6]*(-((1603 + (18980 + (35397*I)*Sqrt[3])^(2/3))/(18980 + (3 
5397*I)*Sqrt[3])^(1/3)) + 18*(-53/18 + x))^(5/2)*(-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)*(((-1/2867157*I)*(18980 + (35397*I) 
*Sqrt[3])^(1/3)*(1603 + (9*(1603 + (18980 + (35397*I)*Sqrt[3])^(2/3))*(-53 
/18 + x))/(18980 + (35397*I)*Sqrt[3])^(1/3)))/(Sqrt[3]*(1603 - (18980 + (3 
5397*I)*Sqrt[3])^(2/3))*(-((1603 + (18980 + (35397*I)*Sqrt[3])^(2/3))/(189 
80 + (35397*I)*Sqrt[3])^(1/3)) + 18*(-53/18 + x))^(3/2)*(-1603 + 2569609/( 
18980 + (35397*I)*Sqrt[3])^(2/3) + (18980 + (35397*I)*Sqrt[3])^(2/3) + (18 
*(1603 + (18980 + (35397*I)*Sqrt[3])^(2/3))*(-53/18 + x))/(18980 + (35397* 
I)*Sqrt[3])^(1/3) + 324*(-53/18 + x)^2)^(3/2)) - (((7*I)/637146)*(18980 + 
(35397*I)*Sqrt[3])^(1/3)*(((-1/955719*I)*(18980 + (35397*I)*Sqrt[3])^(1/3) 
*(1468348 - ((4479323627*I - 671835060*Sqrt[3])*(18980 + (35397*I)*Sqrt[3] 
)^(1/3))/(18980*I - 35397*Sqrt[3]) + (1603*(3038362027*I + 2015505180*Sqrt 
[3]))/((18980*I - 35397*Sqrt[3])*(18980 + (35397*I)*Sqrt[3])^(1/3)) - (18* 
(2569609*I + (18980*I - 35397*Sqrt[3])*(18980 + (35397*I)*Sqrt[3])^(1/3) - 
 (458*I)*(18980 + (35397*I)*Sqrt[3])^(2/3))*(1603 + (18980 + (35397*I)*Sqr 
t[3])^(2/3))*(-53/18 + x))/(18980*I - 35397*Sqrt[3])))/(Sqrt[3]*(1603 - (1 
8980 + (35397*I)*Sqrt[3])^(2/3))*(-((1603 + (18980 + (35397*I)*Sqrt[3])...
 

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 1165

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m + 1)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e) 
*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^ 
2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))   Int[(d 
+ e*x)^m*Simp[b*c*d*e*(2*p - m + 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p 
+ 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x, x]*(a + 
 b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && LtQ[p, -1] 
 && IntQuadraticQ[a, b, c, d, e, m, p, x]
 

rule 1172

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

rule 1235

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

rule 1237

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

rule 1269

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

rule 2475

Int[((a_.) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_Symbol] :> 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/1 
8^(1/3))*x + d^2*x^2, x]^p)   Int[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, x], x]] /; Free 
Q[{a, b, d, p}, x] && NeQ[4*b^3 + 27*a^2*d, 0] &&  !IntegerQ[p]
 

rule 2481

Int[(Px_)^(p_), x_Symbol] :> With[{a = Coeff[Px, x, 0], b = Coeff[Px, x, 1] 
, c = Coeff[Px, x, 2], d = Coeff[Px, x, 3]}, Subst[Int[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, c/(3 
*d) + x]] /; FreeQ[p, x] && PolyQ[Px, x, 3]
 

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.44

method	result	size
risch	\(-\frac {2 \left (17524765440 x^{5}-319649088360 x^{4}[phone] x^{3}-2752459952046 x^{2}-2084092866018 x [phone]\right )}{717822455829363 \left (6 x^{3}-53 x^{2}+67 x +70\right )^{\frac {3}{2}}}+\frac {558049940 \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{313688413197431631 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}+\frac {973598080 \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \left (-\frac {23 \operatorname {EllipticE}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{3}+7 \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )\right )}{313688413197431631 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}\)	\(176\)
default	\(\frac {\left (-\frac {3206}{139216401} x^{2}+\frac {3314}{21981537} x -\frac {13687}{835298406}\right ) \sqrt {6 x^{3}-53 x^{2}+67 x +70}}{\left (x^{3}-\frac {53}{6} x^{2}+\frac {67}{6} x +\frac {35}{3}\right )^{2}}-\frac {12 \left (\frac {13336565795}{2153467367488089}+\frac {486799040}{717822455829363} x^{2}-\frac {723013130}{113340387762531} x \right )}{\sqrt {6 x^{3}-53 x^{2}+67 x +70}}+\frac {558049940 \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{313688413197431631 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}+\frac {973598080 \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \left (-\frac {23 \operatorname {EllipticE}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{3}+7 \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )\right )}{313688413197431631 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}\)	\(204\)
elliptic	\(\frac {\left (-\frac {3206}{139216401} x^{2}+\frac {3314}{21981537} x -\frac {13687}{835298406}\right ) \sqrt {6 x^{3}-53 x^{2}+67 x +70}}{\left (x^{3}-\frac {53}{6} x^{2}+\frac {67}{6} x +\frac {35}{3}\right )^{2}}-\frac {12 \left (\frac {13336565795}{2153467367488089}+\frac {486799040}{717822455829363} x^{2}-\frac {723013130}{113340387762531} x \right )}{\sqrt {6 x^{3}-53 x^{2}+67 x +70}}+\frac {558049940 \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{313688413197431631 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}+\frac {973598080 \sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \left (-\frac {23 \operatorname {EllipticE}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{3}+7 \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )\right )}{313688413197431631 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}\)	\(204\)

Input:

int(1/(6*x^3-53*x^2+67*x+70)^(5/2),x,method=_RETURNVERBOSE)
                                                                                    
                                                                                    
 

Output:

-2/717822455829363*(17524765440*x^5-319649088360*x^4+1811880447440*x^3-275 
2459952046*x^2-2084092866018*x[phone])/(6*x^3-53*x^2+67*x+70)^(3/2) 
[phone]/313688413197431631*(76+114*x)^(1/2)*(483-69*x)^(1/2)*(285-114*x 
)^(1/2)/(6*x^3-53*x^2+67*x+70)^(1/2)*EllipticF(1/19*(76+114*x)^(1/2),1/46* 
874^(1/2))[phone]/313688413197431631*(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+1 
14*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 = 0.11 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.41 \[ \int \frac {1}{\left (70+67 x-53 x^2+6 x^3\right )^{5/2}} \, dx=\frac {2 \, {\left (15411399290 \, \sqrt {6} {\left (36 \, x^{6} - 636 \, x^{5} + 3613 \, x^{4} - 6262 \, x^{3} - 2931 \, x^{2} + 9380 \, x + 4900\right )} {\rm weierstrassPInverse}\left (\frac {1603}{27}, \frac {18980}{729}, x - \frac {53}{18}\right ) - 4381191360 \, \sqrt {6} {\left (36 \, x^{6} - 636 \, x^{5} + 3613 \, x^{4} - 6262 \, x^{3} - 2931 \, x^{2} + 9380 \, x + 4900\right )} {\rm weierstrassZeta}\left (\frac {1603}{27}, \frac {18980}{729}, {\rm weierstrassPInverse}\left (\frac {1603}{27}, \frac {18980}{729}, x - \frac {53}{18}\right )\right ) - 9 \, {\left (17524765440 \, x^{5} - 319649088360 \, x^{4} + 1811880447440 \, x^{3} - 2752459952046 \, x^{2} - 2084092866018 \, x + 2078836420243\right )} \sqrt {6 \, x^{3} - 53 \, x^{2} + 67 \, x + 70}\right )}}{6460402102464267 \, {\left (36 \, x^{6} - 636 \, x^{5} + 3613 \, x^{4} - 6262 \, x^{3} - 2931 \, x^{2} + 9380 \, x + 4900\right )}} \] Input:

integrate(1/(6*x^3-53*x^2+67*x+70)^(5/2),x, algorithm="fricas")
 

Output:

2/6460402102464267*(15411399290*sqrt(6)*(36*x^6 - 636*x^5 + 3613*x^4 - 626 
2*x^3 - 2931*x^2 + 9380*x + 4900)*weierstrassPInverse(1603/27, 18980/729, 
x - 53/18) - 4381191360*sqrt(6)*(36*x^6 - 636*x^5 + 3613*x^4 - 6262*x^3 - 
2931*x^2 + 9380*x + 4900)*weierstrassZeta(1603/27, 18980/729, weierstrassP 
Inverse(1603/27, 18980/729, x - 53/18)) - 9*(17524765440*x^5 - 31964908836 
0*x^4 + 1811880447440*x^3 - 2752459952046*x^2 - 2084092866018*x + 20788364 
20243)*sqrt(6*x^3 - 53*x^2 + 67*x + 70))/(36*x^6 - 636*x^5 + 3613*x^4 - 62 
62*x^3 - 2931*x^2 + 9380*x + 4900)
 

Sympy [F]

\[ \int \frac {1}{\left (70+67 x-53 x^2+6 x^3\right )^{5/2}} \, dx=\int \frac {1}{\left (6 x^{3} - 53 x^{2} + 67 x + 70\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(1/(6*x**3-53*x**2+67*x+70)**(5/2),x)
 

Output:

Integral((6*x**3 - 53*x**2 + 67*x + 70)**(-5/2), x)
 

Maxima [F]

\[ \int \frac {1}{\left (70+67 x-53 x^2+6 x^3\right )^{5/2}} \, dx=\int { \frac {1}{{\left (6 \, x^{3} - 53 \, x^{2} + 67 \, x + 70\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(6*x^3-53*x^2+67*x+70)^(5/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {1}{\left (70+67 x-53 x^2+6 x^3\right )^{5/2}} \, dx=\int { \frac {1}{{\left (6 \, x^{3} - 53 \, x^{2} + 67 \, x + 70\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(6*x^3-53*x^2+67*x+70)^(5/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (70+67 x-53 x^2+6 x^3\right )^{5/2}} \, dx=\int \frac {1}{{\left (6\,x^3-53\,x^2+67\,x+70\right )}^{5/2}} \,d x \] Input:

int(1/(67*x - 53*x^2 + 6*x^3 + 70)^(5/2),x)
 

Output:

int(1/(67*x - 53*x^2 + 6*x^3 + 70)^(5/2), x)
 

Reduce [F]

\[ \int \frac {1}{\left (70+67 x-53 x^2+6 x^3\right )^{5/2}} \, dx=\int \frac {\sqrt {6 x^{3}-53 x^{2}+67 x +70}}{216 x^{9}-5724 x^{8}+57798 x^{7}-269153 x^{6}+511851 x^{5}+44979 x^{4}-1102457 x^{3}+163590 x^{2}+984900 x +343000}d x \] Input:

int(1/(6*x^3-53*x^2+67*x+70)^(5/2),x)
 

Output:

int(sqrt(6*x**3 - 53*x**2 + 67*x + 70)/(216*x**9 - 5724*x**8 + 57798*x**7 
- 269153*x**6 + 511851*x**5 + 44979*x**4 - 1102457*x**3 + 163590*x**2 + 98 
4900*x + 343000),x)