\(\int \frac {\sqrt [3]{a+b x^3}}{x^6 (a d-b d x^3)} \, dx\) [799]

Optimal result

Integrand size = 28, antiderivative size = 523 \[ \int \frac {\sqrt [3]{a+b x^3}}{x^6 \left (a d-b d x^3\right )} \, dx=-\frac {\sqrt [3]{a+b x^3}}{5 a d x^5}-\frac {3 b \sqrt [3]{a+b x^3}}{5 a^2 d x^2}-\frac {\sqrt [3]{2} b^{5/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} a^{7/3} d}-\frac {b^{5/3} \arctan \left (\frac {1+\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3} a^{7/3} d}+\frac {2 b^2 x \left (1+\frac {b x^3}{a}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{5 a^2 d \left (a+b x^3\right )^{2/3}}-\frac {b^{5/3} \log \left (2^{2/3}-\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3\ 2^{2/3} a^{7/3} d}+\frac {b^{5/3} \log \left (1+\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}\right )}{3\ 2^{2/3} a^{7/3} d}-\frac {\sqrt [3]{2} b^{5/3} \log \left (1+\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}\right )}{3 a^{7/3} d}+\frac {b^{5/3} \log \left (2 \sqrt [3]{2}+\frac {\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}\right )}{6\ 2^{2/3} a^{7/3} d} \] Output:

-1/5*(b*x^3+a)^(1/3)/a/d/x^5-3/5*b*(b*x^3+a)^(1/3)/a^2/d/x^2-1/3*2^(1/3)*b 
^(5/3)*arctan(1/3*(1-2*2^(1/3)*(a^(1/3)+b^(1/3)*x)/(b*x^3+a)^(1/3))*3^(1/2 
))*3^(1/2)/a^(7/3)/d-1/6*b^(5/3)*arctan(1/3*(1+2^(1/3)*(a^(1/3)+b^(1/3)*x) 
/(b*x^3+a)^(1/3))*3^(1/2))*2^(1/3)*3^(1/2)/a^(7/3)/d+2/5*b^2*x*(1+b*x^3/a) 
^(2/3)*hypergeom([1/3, 2/3],[4/3],-b*x^3/a)/a^2/d/(b*x^3+a)^(2/3)-1/6*b^(5 
/3)*ln(2^(2/3)-(a^(1/3)+b^(1/3)*x)/(b*x^3+a)^(1/3))*2^(1/3)/a^(7/3)/d+1/6* 
b^(5/3)*ln(1+2^(2/3)*(a^(1/3)+b^(1/3)*x)^2/(b*x^3+a)^(2/3)-2^(1/3)*(a^(1/3 
)+b^(1/3)*x)/(b*x^3+a)^(1/3))*2^(1/3)/a^(7/3)/d-1/3*2^(1/3)*b^(5/3)*ln(1+2 
^(1/3)*(a^(1/3)+b^(1/3)*x)/(b*x^3+a)^(1/3))/a^(7/3)/d+1/12*b^(5/3)*ln(2*2^ 
(1/3)+(a^(1/3)+b^(1/3)*x)^2/(b*x^3+a)^(2/3)+2^(2/3)*(a^(1/3)+b^(1/3)*x)/(b 
*x^3+a)^(1/3))*2^(1/3)/a^(7/3)/d
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.

Time = 11.15 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.46 \[ \int \frac {\sqrt [3]{a+b x^3}}{x^6 \left (a d-b d x^3\right )} \, dx=\frac {-\frac {4 \left (a^2+4 a b x^3+3 b^2 x^6\right )}{a^2 x^5}+\frac {3 b^3 x^4 \left (1+\frac {b x^3}{a}\right )^{2/3} \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},1,\frac {7}{3},-\frac {b x^3}{a},\frac {b x^3}{a}\right )}{a^3}+\frac {112 b^2 x \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},1,\frac {4}{3},-\frac {b x^3}{a},\frac {b x^3}{a}\right )}{\left (a-b x^3\right ) \left (4 a \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},1,\frac {4}{3},-\frac {b x^3}{a},\frac {b x^3}{a}\right )+b x^3 \left (3 \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},2,\frac {7}{3},-\frac {b x^3}{a},\frac {b x^3}{a}\right )-2 \operatorname {AppellF1}\left (\frac {4}{3},\frac {5}{3},1,\frac {7}{3},-\frac {b x^3}{a},\frac {b x^3}{a}\right )\right )\right )}}{20 d \left (a+b x^3\right )^{2/3}} \] Input:

Integrate[(a + b*x^3)^(1/3)/(x^6*(a*d - b*d*x^3)),x]
 

Output:

((-4*(a^2 + 4*a*b*x^3 + 3*b^2*x^6))/(a^2*x^5) + (3*b^3*x^4*(1 + (b*x^3)/a) 
^(2/3)*AppellF1[4/3, 2/3, 1, 7/3, -((b*x^3)/a), (b*x^3)/a])/a^3 + (112*b^2 
*x*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), (b*x^3)/a])/((a - b*x^3)*(4*a* 
AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), (b*x^3)/a] + b*x^3*(3*AppellF1[4/ 
3, 2/3, 2, 7/3, -((b*x^3)/a), (b*x^3)/a] - 2*AppellF1[4/3, 5/3, 1, 7/3, -( 
(b*x^3)/a), (b*x^3)/a]))))/(20*d*(a + b*x^3)^(2/3))
 

Rubi [A] (verified)

Time = 1.39 (sec) , antiderivative size = 581, normalized size of antiderivative = 1.11, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {975, 27, 1053, 25, 27, 1026, 779, 778, 928, 779, 778, 927, 982, 821, 16, 1142, 25, 27, 1082, 217, 1103}

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.

\(\displaystyle \int \frac {\sqrt [3]{a+b x^3}}{x^6 \left (a d-b d x^3\right )} \, dx\)

\(\Big \downarrow \) 975

\(\displaystyle \frac {\int \frac {2 b \left (2 b x^3+3 a\right )}{x^3 \left (a-b x^3\right ) \left (b x^3+a\right )^{2/3}}dx}{5 a d}-\frac {\sqrt [3]{a+b x^3}}{5 a d x^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 b \int \frac {2 b x^3+3 a}{x^3 \left (a-b x^3\right ) \left (b x^3+a\right )^{2/3}}dx}{5 a d}-\frac {\sqrt [3]{a+b x^3}}{5 a d x^5}\)

\(\Big \downarrow \) 1053

\(\displaystyle \frac {2 b \left (-\frac {\int -\frac {a b \left (3 b x^3+7 a\right )}{\left (a-b x^3\right ) \left (b x^3+a\right )^{2/3}}dx}{2 a^2}-\frac {3 \sqrt [3]{a+b x^3}}{2 a x^2}\right )}{5 a d}-\frac {\sqrt [3]{a+b x^3}}{5 a d x^5}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 b \left (\frac {\int \frac {a b \left (3 b x^3+7 a\right )}{\left (a-b x^3\right ) \left (b x^3+a\right )^{2/3}}dx}{2 a^2}-\frac {3 \sqrt [3]{a+b x^3}}{2 a x^2}\right )}{5 a d}-\frac {\sqrt [3]{a+b x^3}}{5 a d x^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 b \left (\frac {b \int \frac {3 b x^3+7 a}{\left (a-b x^3\right ) \left (b x^3+a\right )^{2/3}}dx}{2 a}-\frac {3 \sqrt [3]{a+b x^3}}{2 a x^2}\right )}{5 a d}-\frac {\sqrt [3]{a+b x^3}}{5 a d x^5}\)

\(\Big \downarrow \) 1026

\(\displaystyle \frac {2 b \left (\frac {b \left (10 a \int \frac {1}{\left (a-b x^3\right ) \left (b x^3+a\right )^{2/3}}dx-3 \int \frac {1}{\left (b x^3+a\right )^{2/3}}dx\right )}{2 a}-\frac {3 \sqrt [3]{a+b x^3}}{2 a x^2}\right )}{5 a d}-\frac {\sqrt [3]{a+b x^3}}{5 a d x^5}\)

\(\Big \downarrow \) 779

\(\displaystyle \frac {2 b \left (\frac {b \left (10 a \int \frac {1}{\left (a-b x^3\right ) \left (b x^3+a\right )^{2/3}}dx-\frac {3 \left (\frac {b x^3}{a}+1\right )^{2/3} \int \frac {1}{\left (\frac {b x^3}{a}+1\right )^{2/3}}dx}{\left (a+b x^3\right )^{2/3}}\right )}{2 a}-\frac {3 \sqrt [3]{a+b x^3}}{2 a x^2}\right )}{5 a d}-\frac {\sqrt [3]{a+b x^3}}{5 a d x^5}\)

\(\Big \downarrow \) 778

\(\displaystyle \frac {2 b \left (\frac {b \left (10 a \int \frac {1}{\left (a-b x^3\right ) \left (b x^3+a\right )^{2/3}}dx-\frac {3 x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}\right )}{2 a}-\frac {3 \sqrt [3]{a+b x^3}}{2 a x^2}\right )}{5 a d}-\frac {\sqrt [3]{a+b x^3}}{5 a d x^5}\)

\(\Big \downarrow \) 928

\(\displaystyle \frac {2 b \left (\frac {b \left (10 a \left (\frac {\int \frac {1}{\left (b x^3+a\right )^{2/3}}dx}{2 a}+\frac {\int \frac {\sqrt [3]{b x^3+a}}{a-b x^3}dx}{2 a}\right )-\frac {3 x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}\right )}{2 a}-\frac {3 \sqrt [3]{a+b x^3}}{2 a x^2}\right )}{5 a d}-\frac {\sqrt [3]{a+b x^3}}{5 a d x^5}\)

\(\Big \downarrow \) 779

\(\displaystyle \frac {2 b \left (\frac {b \left (10 a \left (\frac {\int \frac {\sqrt [3]{b x^3+a}}{a-b x^3}dx}{2 a}+\frac {\left (\frac {b x^3}{a}+1\right )^{2/3} \int \frac {1}{\left (\frac {b x^3}{a}+1\right )^{2/3}}dx}{2 a \left (a+b x^3\right )^{2/3}}\right )-\frac {3 x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}\right )}{2 a}-\frac {3 \sqrt [3]{a+b x^3}}{2 a x^2}\right )}{5 a d}-\frac {\sqrt [3]{a+b x^3}}{5 a d x^5}\)

\(\Big \downarrow \) 778

\(\displaystyle \frac {2 b \left (\frac {b \left (10 a \left (\frac {\int \frac {\sqrt [3]{b x^3+a}}{a-b x^3}dx}{2 a}+\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{2 a \left (a+b x^3\right )^{2/3}}\right )-\frac {3 x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}\right )}{2 a}-\frac {3 \sqrt [3]{a+b x^3}}{2 a x^2}\right )}{5 a d}-\frac {\sqrt [3]{a+b x^3}}{5 a d x^5}\)

\(\Big \downarrow \) 927

\(\displaystyle \frac {2 b \left (\frac {b \left (10 a \left (\frac {9 \int \frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a} \left (4-\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^3}{b x^3+a}\right ) \left (\frac {2 \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^3}{b x^3+a}+1\right )}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{2 a^{2/3} \sqrt [3]{b}}+\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{2 a \left (a+b x^3\right )^{2/3}}\right )-\frac {3 x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}\right )}{2 a}-\frac {3 \sqrt [3]{a+b x^3}}{2 a x^2}\right )}{5 a d}-\frac {\sqrt [3]{a+b x^3}}{5 a d x^5}\)

\(\Big \downarrow \) 982

\(\displaystyle \frac {2 b \left (\frac {b \left (10 a \left (\frac {9 \left (\frac {1}{9} \int \frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a} \left (4-\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^3}{b x^3+a}\right )}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}+\frac {2}{9} \int \frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a} \left (\frac {2 \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^3}{b x^3+a}+1\right )}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}\right )}{2 a^{2/3} \sqrt [3]{b}}+\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{2 a \left (a+b x^3\right )^{2/3}}\right )-\frac {3 x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}\right )}{2 a}-\frac {3 \sqrt [3]{a+b x^3}}{2 a x^2}\right )}{5 a d}-\frac {\sqrt [3]{a+b x^3}}{5 a d x^5}\)

\(\Big \downarrow \) 821

\(\displaystyle \frac {2 b \left (\frac {b \left (10 a \left (\frac {9 \left (\frac {2}{9} \left (\frac {\int \frac {\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}{\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{3 \sqrt [3]{2} \sqrt [3]{a}}-\frac {\int \frac {1}{\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{3 \sqrt [3]{2} \sqrt [3]{a}}\right )+\frac {1}{9} \left (\frac {\int \frac {1}{2^{2/3}-\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{b x^3+a}}}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{3\ 2^{2/3} \sqrt [3]{a}}-\frac {\int \frac {2^{2/3}-\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{b x^3+a}}}{\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+2 \sqrt [3]{2}}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{3\ 2^{2/3} \sqrt [3]{a}}\right )\right )}{2 a^{2/3} \sqrt [3]{b}}+\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{2 a \left (a+b x^3\right )^{2/3}}\right )-\frac {3 x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}\right )}{2 a}-\frac {3 \sqrt [3]{a+b x^3}}{2 a x^2}\right )}{5 a d}-\frac {\sqrt [3]{a+b x^3}}{5 a d x^5}\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {2 b \left (\frac {b \left (10 a \left (\frac {9 \left (\frac {2}{9} \left (\frac {\int \frac {\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}{\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{3 \sqrt [3]{2} \sqrt [3]{a}}-\frac {\log \left (\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3\ 2^{2/3} a^{2/3}}\right )+\frac {1}{9} \left (-\frac {\int \frac {2^{2/3}-\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{b x^3+a}}}{\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+2 \sqrt [3]{2}}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{3\ 2^{2/3} \sqrt [3]{a}}-\frac {\log \left (2^{2/3}-\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3\ 2^{2/3} a^{2/3}}\right )\right )}{2 a^{2/3} \sqrt [3]{b}}+\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{2 a \left (a+b x^3\right )^{2/3}}\right )-\frac {3 x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}\right )}{2 a}-\frac {3 \sqrt [3]{a+b x^3}}{2 a x^2}\right )}{5 a d}-\frac {\sqrt [3]{a+b x^3}}{5 a d x^5}\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {2 b \left (\frac {b \left (10 a \left (\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{2 a \left (b x^3+a\right )^{2/3}}+\frac {9 \left (\frac {2}{9} \left (\frac {\frac {3}{2} \int \frac {1}{\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}+\frac {\int -\frac {\sqrt [3]{2} \sqrt [3]{a} \left (1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}\right )}{\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{2 \sqrt [3]{2} \sqrt [3]{a}}}{3 \sqrt [3]{2} \sqrt [3]{a}}-\frac {\log \left (\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1\right )}{3\ 2^{2/3} a^{2/3}}\right )+\frac {1}{9} \left (-\frac {\log \left (2^{2/3}-\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{b x^3+a}}\right )}{3\ 2^{2/3} a^{2/3}}-\frac {\frac {3 \int \frac {1}{\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+2 \sqrt [3]{2}}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{\sqrt [3]{2}}-\frac {\int \frac {2^{2/3} \sqrt [3]{a} \left (\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1\right )}{\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+2 \sqrt [3]{2}}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{2 \sqrt [3]{a}}}{3\ 2^{2/3} \sqrt [3]{a}}\right )\right )}{2 a^{2/3} \sqrt [3]{b}}\right )-\frac {3 x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{\left (b x^3+a\right )^{2/3}}\right )}{2 a}-\frac {3 \sqrt [3]{b x^3+a}}{2 a x^2}\right )}{5 a d}-\frac {\sqrt [3]{b x^3+a}}{5 a d x^5}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 b \left (\frac {b \left (10 a \left (\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{2 a \left (b x^3+a\right )^{2/3}}+\frac {9 \left (\frac {2}{9} \left (\frac {\frac {3}{2} \int \frac {1}{\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}-\frac {\int \frac {\sqrt [3]{2} \sqrt [3]{a} \left (1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}\right )}{\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{2 \sqrt [3]{2} \sqrt [3]{a}}}{3 \sqrt [3]{2} \sqrt [3]{a}}-\frac {\log \left (\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1\right )}{3\ 2^{2/3} a^{2/3}}\right )+\frac {1}{9} \left (-\frac {\log \left (2^{2/3}-\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{b x^3+a}}\right )}{3\ 2^{2/3} a^{2/3}}-\frac {\frac {3 \int \frac {1}{\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+2 \sqrt [3]{2}}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{\sqrt [3]{2}}-\frac {\int \frac {2^{2/3} \sqrt [3]{a} \left (\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1\right )}{\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+2 \sqrt [3]{2}}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{2 \sqrt [3]{a}}}{3\ 2^{2/3} \sqrt [3]{a}}\right )\right )}{2 a^{2/3} \sqrt [3]{b}}\right )-\frac {3 x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{\left (b x^3+a\right )^{2/3}}\right )}{2 a}-\frac {3 \sqrt [3]{b x^3+a}}{2 a x^2}\right )}{5 a d}-\frac {\sqrt [3]{b x^3+a}}{5 a d x^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 b \left (\frac {b \left (10 a \left (\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{2 a \left (b x^3+a\right )^{2/3}}+\frac {9 \left (\frac {2}{9} \left (\frac {\frac {3}{2} \int \frac {1}{\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}-\frac {1}{2} \int \frac {1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}}{\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{3 \sqrt [3]{2} \sqrt [3]{a}}-\frac {\log \left (\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1\right )}{3\ 2^{2/3} a^{2/3}}\right )+\frac {1}{9} \left (-\frac {\log \left (2^{2/3}-\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{b x^3+a}}\right )}{3\ 2^{2/3} a^{2/3}}-\frac {\frac {3 \int \frac {1}{\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+2 \sqrt [3]{2}}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{\sqrt [3]{2}}-\frac {\int \frac {\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}{\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+2 \sqrt [3]{2}}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{\sqrt [3]{2}}}{3\ 2^{2/3} \sqrt [3]{a}}\right )\right )}{2 a^{2/3} \sqrt [3]{b}}\right )-\frac {3 x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{\left (b x^3+a\right )^{2/3}}\right )}{2 a}-\frac {3 \sqrt [3]{b x^3+a}}{2 a x^2}\right )}{5 a d}-\frac {\sqrt [3]{b x^3+a}}{5 a d x^5}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {2 b \left (\frac {b \left (10 a \left (\frac {9 \left (\frac {2}{9} \left (\frac {\frac {3 \int \frac {1}{-\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{a^{2/3} \left (b x^3+a\right )^{2/3}}-3}d\left (1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}\right )}{\sqrt [3]{2} \sqrt [3]{a}}-\frac {1}{2} \int \frac {1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}}{\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{3 \sqrt [3]{2} \sqrt [3]{a}}-\frac {\log \left (\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3\ 2^{2/3} a^{2/3}}\right )+\frac {1}{9} \left (-\frac {-\frac {3 \int \frac {1}{-\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{a^{2/3} \left (b x^3+a\right )^{2/3}}-3}d\left (\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1\right )}{\sqrt [3]{a}}-\frac {\int \frac {\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}{\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+2 \sqrt [3]{2}}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{\sqrt [3]{2}}}{3\ 2^{2/3} \sqrt [3]{a}}-\frac {\log \left (2^{2/3}-\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3\ 2^{2/3} a^{2/3}}\right )\right )}{2 a^{2/3} \sqrt [3]{b}}+\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{2 a \left (a+b x^3\right )^{2/3}}\right )-\frac {3 x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}\right )}{2 a}-\frac {3 \sqrt [3]{a+b x^3}}{2 a x^2}\right )}{5 a d}-\frac {\sqrt [3]{a+b x^3}}{5 a d x^5}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {2 b \left (\frac {b \left (10 a \left (\frac {9 \left (\frac {2}{9} \left (\frac {-\frac {1}{2} \int \frac {1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}}{\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{a}}}{3 \sqrt [3]{2} \sqrt [3]{a}}-\frac {\log \left (\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3\ 2^{2/3} a^{2/3}}\right )+\frac {1}{9} \left (-\frac {\frac {\sqrt {3} \arctan \left (\frac {\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt [3]{a}}-\frac {\int \frac {\frac {\sqrt [3]{2} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+1}{\frac {\left (\sqrt [3]{b} x+\sqrt [3]{a}\right )^2}{\left (b x^3+a\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}{\sqrt [3]{b x^3+a}}+2 \sqrt [3]{2}}d\frac {\sqrt [3]{b} x+\sqrt [3]{a}}{\sqrt [3]{a} \sqrt [3]{b x^3+a}}}{\sqrt [3]{2}}}{3\ 2^{2/3} \sqrt [3]{a}}-\frac {\log \left (2^{2/3}-\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3\ 2^{2/3} a^{2/3}}\right )\right )}{2 a^{2/3} \sqrt [3]{b}}+\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{2 a \left (a+b x^3\right )^{2/3}}\right )-\frac {3 x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}\right )}{2 a}-\frac {3 \sqrt [3]{a+b x^3}}{2 a x^2}\right )}{5 a d}-\frac {\sqrt [3]{a+b x^3}}{5 a d x^5}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {2 b \left (\frac {b \left (10 a \left (\frac {9 \left (\frac {2}{9} \left (\frac {\frac {\log \left (\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{2 \sqrt [3]{2} \sqrt [3]{a}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt [3]{a}}}{3 \sqrt [3]{2} \sqrt [3]{a}}-\frac {\log \left (\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3\ 2^{2/3} a^{2/3}}\right )+\frac {1}{9} \left (-\frac {\log \left (2^{2/3}-\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3\ 2^{2/3} a^{2/3}}-\frac {\frac {\sqrt {3} \arctan \left (\frac {\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt [3]{a}}-\frac {\log \left (\frac {\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+2 \sqrt [3]{2}\right )}{2 \sqrt [3]{a}}}{3\ 2^{2/3} \sqrt [3]{a}}\right )\right )}{2 a^{2/3} \sqrt [3]{b}}+\frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{2 a \left (a+b x^3\right )^{2/3}}\right )-\frac {3 x \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}}\right )}{2 a}-\frac {3 \sqrt [3]{a+b x^3}}{2 a x^2}\right )}{5 a d}-\frac {\sqrt [3]{a+b x^3}}{5 a d x^5}\)

Input:

Int[(a + b*x^3)^(1/3)/(x^6*(a*d - b*d*x^3)),x]
 

Output:

-1/5*(a + b*x^3)^(1/3)/(a*d*x^5) + (2*b*((-3*(a + b*x^3)^(1/3))/(2*a*x^2) 
+ (b*((-3*x*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, -((b*x^ 
3)/a)])/(a + b*x^3)^(2/3) + 10*a*((x*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2 
F1[1/3, 2/3, 4/3, -((b*x^3)/a)])/(2*a*(a + b*x^3)^(2/3)) + (9*((2*((-((Sqr 
t[3]*ArcTan[(1 - (2*2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3))/Sqrt 
[3]])/(2^(1/3)*a^(1/3))) + Log[1 + (2^(2/3)*(a^(1/3) + b^(1/3)*x)^2)/(a + 
b*x^3)^(2/3) - (2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3)]/(2*2^(1/ 
3)*a^(1/3)))/(3*2^(1/3)*a^(1/3)) - Log[1 + (2^(1/3)*(a^(1/3) + b^(1/3)*x)) 
/(a + b*x^3)^(1/3)]/(3*2^(2/3)*a^(2/3))))/9 + (-1/3*Log[2^(2/3) - (a^(1/3) 
 + b^(1/3)*x)/(a + b*x^3)^(1/3)]/(2^(2/3)*a^(2/3)) - ((Sqrt[3]*ArcTan[(1 + 
 (2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3))/Sqrt[3]])/a^(1/3) - Lo 
g[2*2^(1/3) + (a^(1/3) + b^(1/3)*x)^2/(a + b*x^3)^(2/3) + (2^(2/3)*(a^(1/3 
) + b^(1/3)*x))/(a + b*x^3)^(1/3)]/(2*a^(1/3)))/(3*2^(2/3)*a^(1/3)))/9))/( 
2*a^(2/3)*b^(1/3)))))/(2*a)))/(5*a*d)
 

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 778

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F 
1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p 
, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p] || 
GtQ[a, 0])
 

rule 779

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x 
^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p])   Int[(1 + b*(x^n/a))^p, x], x 
] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Si 
mplify[1/n + p], 0] &&  !(IntegerQ[p] || GtQ[a, 0])
 

rule 821

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Simp[-(3*Rt[a, 3]*Rt[b, 3])^(- 
1)   Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]*Rt[b, 3]) 
 Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2 
*x^2), x], x] /; FreeQ[{a, b}, x]
 

rule 927

Int[((a_) + (b_.)*(x_)^3)^(1/3)/((c_) + (d_.)*(x_)^3), x_Symbol] :> With[{q 
 = Rt[b/a, 3]}, Simp[9*(a/(c*q))   Subst[Int[x/((4 - a*x^3)*(1 + 2*a*x^3)), 
 x], x, (1 + q*x)/(a + b*x^3)^(1/3)], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[ 
b*c - a*d, 0] && EqQ[b*c + a*d, 0]
 

rule 928

Int[1/(((a_) + (b_.)*(x_)^3)^(2/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> Sim 
p[b/(b*c - a*d)   Int[1/(a + b*x^3)^(2/3), x], x] - Simp[d/(b*c - a*d)   In 
t[(a + b*x^3)^(1/3)/(c + d*x^3), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b* 
c - a*d, 0] && EqQ[b*c + a*d, 0]
 

rule 975

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_) 
)^(q_), x_Symbol] :> Simp[(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/ 
(a*e*(m + 1))), x] - Simp[1/(a*e^n*(m + 1))   Int[(e*x)^(m + n)*(a + b*x^n) 
^p*(c + d*x^n)^(q - 1)*Simp[c*b*(m + 1) + n*(b*c*(p + 1) + a*d*q) + d*(b*(m 
 + 1) + b*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && 
 NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[0, q, 1] && LtQ[m, -1] && IntBinomi 
alQ[a, b, c, d, e, m, n, p, q, x]
 

rule 982

Int[((e_.)*(x_))^(m_.)/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), 
 x_Symbol] :> Simp[b/(b*c - a*d)   Int[(e*x)^m/(a + b*x^n), x], x] - Simp[d 
/(b*c - a*d)   Int[(e*x)^m/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e, m}, 
 x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0]
 

rule 1026

Int[(((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)* 
(x_)^(n_)), x_Symbol] :> Simp[f/d   Int[(a + b*x^n)^p, x], x] + Simp[(d*e - 
 c*f)/d   Int[(a + b*x^n)^p/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, 
 p, n}, x]
 

rule 1053

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

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1142

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[(2*c*d - b*e)/(2*c)   Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) 
Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
 

Maple [F]

\[\int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x^{6} \left (-b d \,x^{3}+a d \right )}d x\]

Input:

int((b*x^3+a)^(1/3)/x^6/(-b*d*x^3+a*d),x)
 

Output:

int((b*x^3+a)^(1/3)/x^6/(-b*d*x^3+a*d),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt [3]{a+b x^3}}{x^6 \left (a d-b d x^3\right )} \, dx=\text {Timed out} \] Input:

integrate((b*x^3+a)^(1/3)/x^6/(-b*d*x^3+a*d),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\sqrt [3]{a+b x^3}}{x^6 \left (a d-b d x^3\right )} \, dx=- \frac {\int \frac {\sqrt [3]{a + b x^{3}}}{- a x^{6} + b x^{9}}\, dx}{d} \] Input:

integrate((b*x**3+a)**(1/3)/x**6/(-b*d*x**3+a*d),x)
 

Output:

-Integral((a + b*x**3)**(1/3)/(-a*x**6 + b*x**9), x)/d
 

Maxima [F]

\[ \int \frac {\sqrt [3]{a+b x^3}}{x^6 \left (a d-b d x^3\right )} \, dx=\int { -\frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{{\left (b d x^{3} - a d\right )} x^{6}} \,d x } \] Input:

integrate((b*x^3+a)^(1/3)/x^6/(-b*d*x^3+a*d),x, algorithm="maxima")
 

Output:

-integrate((b*x^3 + a)^(1/3)/((b*d*x^3 - a*d)*x^6), x)
 

Giac [F]

\[ \int \frac {\sqrt [3]{a+b x^3}}{x^6 \left (a d-b d x^3\right )} \, dx=\int { -\frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{{\left (b d x^{3} - a d\right )} x^{6}} \,d x } \] Input:

integrate((b*x^3+a)^(1/3)/x^6/(-b*d*x^3+a*d),x, algorithm="giac")
 

Output:

integrate(-(b*x^3 + a)^(1/3)/((b*d*x^3 - a*d)*x^6), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt [3]{a+b x^3}}{x^6 \left (a d-b d x^3\right )} \, dx=\int \frac {{\left (b\,x^3+a\right )}^{1/3}}{x^6\,\left (a\,d-b\,d\,x^3\right )} \,d x \] Input:

int((a + b*x^3)^(1/3)/(x^6*(a*d - b*d*x^3)),x)
 

Output:

int((a + b*x^3)^(1/3)/(x^6*(a*d - b*d*x^3)), x)
 

Reduce [F]

\[ \int \frac {\sqrt [3]{a+b x^3}}{x^6 \left (a d-b d x^3\right )} \, dx=\frac {\int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{-b \,x^{9}+a \,x^{6}}d x}{d} \] Input:

int((b*x^3+a)^(1/3)/x^6/(-b*d*x^3+a*d),x)
 

Output:

int((a + b*x**3)**(1/3)/(a*x**6 - b*x**9),x)/d