\(\int \frac {x (a+b x^3)^{2/3}}{a d-b d x^3} \, dx\) [816]

Optimal result

Integrand size = 26, antiderivative size = 457 \[ \int \frac {x \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx=\frac {2^{2/3} \sqrt [3]{a} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} b^{2/3} d}+\frac {\sqrt [3]{a} \arctan \left (\frac {1+\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3} b^{2/3} d}-\frac {x^2 \sqrt [3]{1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},-\frac {b x^3}{a}\right )}{2 d \sqrt [3]{a+b x^3}}+\frac {\sqrt [3]{a} \log \left (\frac {\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a}\right )}{6 \sqrt [3]{2} b^{2/3} d}+\frac {\sqrt [3]{a} \log \left (1+\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}\right )}{3 \sqrt [3]{2} b^{2/3} d}-\frac {2^{2/3} \sqrt [3]{a} \log \left (1+\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}\right )}{3 b^{2/3} d}-\frac {\sqrt [3]{a} \log \left (\frac {\sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a}}-\frac {2^{2/3} \sqrt [3]{b} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}\right )}{2 \sqrt [3]{2} b^{2/3} d} \] Output:

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

Mathematica [C] (verified)

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

Time = 9.70 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.14 \[ \int \frac {x \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx=\frac {x^2 \sqrt [3]{1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {2}{3},-\frac {2}{3},1,\frac {5}{3},-\frac {b x^3}{a},\frac {b x^3}{a}\right )}{2 d \sqrt [3]{a+b x^3}} \] Input:

Integrate[(x*(a + b*x^3)^(2/3))/(a*d - b*d*x^3),x]
 

Output:

(x^2*(1 + (b*x^3)/a)^(1/3)*AppellF1[2/3, -2/3, 1, 5/3, -((b*x^3)/a), (b*x^ 
3)/a])/(2*d*(a + b*x^3)^(1/3))
 

Rubi [A] (verified)

Time = 1.18 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.01, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {984, 27, 889, 888, 991, 27, 750, 16, 27, 1142, 25, 27, 1082, 217, 1103, 2574}

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.

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

\(\Big \downarrow \) 984

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 889

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

\(\Big \downarrow \) 888

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

\(\Big \downarrow \) 991

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 750

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1142

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1103

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

\(\Big \downarrow \) 2574

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

Input:

Int[(x*(a + b*x^3)^(2/3))/(a*d - b*d*x^3),x]
 

Output:

-1/2*(x^2*(1 + (b*x^3)/a)^(1/3)*Hypergeometric2F1[1/3, 2/3, 5/3, -((b*x^3) 
/a)])/(d*(a + b*x^3)^(1/3)) + (2*a*(-(((2^(1/3)*(-((Sqrt[3]*ArcTan[(1 - (2 
*2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3))/Sqrt[3]])/(2^(2/3)*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^(2/3)*a^(1/3))))/3 + L 
og[1 + (2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3)]/(3*2^(1/3)*a^(1/ 
3)))/(a^(1/3)*b^(2/3))) + ((Sqrt[3]*ArcTan[(1 + (2^(1/3)*(a^(1/3) + b^(1/3 
)*x))/(a + b*x^3)^(1/3))/Sqrt[3]])/(2*2^(1/3)*a^(1/3)*b^(1/3)) + Log[(a^(1 
/3) - b^(1/3)*x)^2*(a^(1/3) + b^(1/3)*x)]/(4*2^(1/3)*a^(1/3)*b^(1/3)) - (3 
*Log[b^(1/3)*(a^(1/3) + b^(1/3)*x) - 2^(2/3)*b^(1/3)*(a + b*x^3)^(1/3)])/( 
4*2^(1/3)*a^(1/3)*b^(1/3)))/(3*a^(1/3)*b^(1/3))))/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 750

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2)   Int[1/ 
(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]^2)   Int[(2*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 888

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p 
*((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 
, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] && (ILt 
Q[p, 0] || GtQ[a, 0])
 

rule 889

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^I 
ntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p])   Int[(c*x) 
^m*(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0 
] &&  !(ILtQ[p, 0] || GtQ[a, 0])
 

rule 984

Int[((x_)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^(n_)), x_Symbol 
] :> Simp[b/d   Int[x*(a + b*x^n)^(p - 1), x], x] - Simp[(b*c - a*d)/d   In 
t[x*((a + b*x^n)^(p - 1)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d}, x] && 
NeQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[p, 0] && IntBinomialQ[a, b, c, d, 1, 
 1, n, p, -1, x]
 

rule 991

Int[(x_)/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> 
With[{q = Rt[b/a, 3]}, Simp[-q^2/(3*d)   Int[1/((1 - q*x)*(a + b*x^3)^(1/3) 
), x], x] + Simp[q/d   Subst[Int[1/(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 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]
 

rule 2574

Int[1/(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^3)^(1/3)), x_Symbol] :> Simp[ 
Sqrt[3]*(ArcTan[(1 - 2^(1/3)*Rt[b, 3]*((c - d*x)/(d*(a + b*x^3)^(1/3))))/Sq 
rt[3]]/(2^(4/3)*Rt[b, 3]*c)), x] + (Simp[Log[(c + d*x)^2*(c - d*x)]/(2^(7/3 
)*Rt[b, 3]*c), x] - Simp[(3*Log[Rt[b, 3]*(c - d*x) + 2^(2/3)*d*(a + b*x^3)^ 
(1/3)])/(2^(7/3)*Rt[b, 3]*c), x]) /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^3 + 
a*d^3, 0]
 

Maple [F]

\[\int \frac {x \left (b \,x^{3}+a \right )^{\frac {2}{3}}}{-b d \,x^{3}+a d}d x\]

Input:

int(x*(b*x^3+a)^(2/3)/(-b*d*x^3+a*d),x)
 

Output:

int(x*(b*x^3+a)^(2/3)/(-b*d*x^3+a*d),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {x \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx=\text {Timed out} \] Input:

integrate(x*(b*x^3+a)^(2/3)/(-b*d*x^3+a*d),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {x \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx=- \frac {\int \frac {x \left (a + b x^{3}\right )^{\frac {2}{3}}}{- a + b x^{3}}\, dx}{d} \] Input:

integrate(x*(b*x**3+a)**(2/3)/(-b*d*x**3+a*d),x)
                                                                                    
                                                                                    
 

Output:

-Integral(x*(a + b*x**3)**(2/3)/(-a + b*x**3), x)/d
 

Maxima [F]

\[ \int \frac {x \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx=\int { -\frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}} x}{b d x^{3} - a d} \,d x } \] Input:

integrate(x*(b*x^3+a)^(2/3)/(-b*d*x^3+a*d),x, algorithm="maxima")
 

Output:

-integrate((b*x^3 + a)^(2/3)*x/(b*d*x^3 - a*d), x)
 

Giac [F]

\[ \int \frac {x \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx=\int { -\frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}} x}{b d x^{3} - a d} \,d x } \] Input:

integrate(x*(b*x^3+a)^(2/3)/(-b*d*x^3+a*d),x, algorithm="giac")
 

Output:

integrate(-(b*x^3 + a)^(2/3)*x/(b*d*x^3 - a*d), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx=\int \frac {x\,{\left (b\,x^3+a\right )}^{2/3}}{a\,d-b\,d\,x^3} \,d x \] Input:

int((x*(a + b*x^3)^(2/3))/(a*d - b*d*x^3),x)
 

Output:

int((x*(a + b*x^3)^(2/3))/(a*d - b*d*x^3), x)
 

Reduce [F]

\[ \int \frac {x \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx=\frac {\int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}} x}{-b \,x^{3}+a}d x}{d} \] Input:

int(x*(b*x^3+a)^(2/3)/(-b*d*x^3+a*d),x)
 

Output:

int(((a + b*x**3)**(2/3)*x)/(a - b*x**3),x)/d