\(\int x (a+b \log (c (d+\frac {e}{x^{2/3}})^n))^3 \, dx\) [525]

Optimal result

Integrand size = 22, antiderivative size = 451 \[ \int x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx=\frac {3 b^2 e^2 n^2 \left (d+\frac {e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{2 d^3}+\frac {3 b^2 e^3 n^2 \log \left (1-\frac {d}{d+\frac {e}{x^{2/3}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{2 d^3}-\frac {3 b e^2 n \left (d+\frac {e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{2 d^3}+\frac {3 b e n x^{4/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{4 d}-\frac {3 b e^3 n \log \left (1-\frac {d}{d+\frac {e}{x^{2/3}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{2 d^3}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3+\frac {3 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \log \left (-\frac {e}{d x^{2/3}}\right )}{d^3}+\frac {b^3 e^3 n^3 \log (x)}{d^3}-\frac {3 b^3 e^3 n^3 \operatorname {PolyLog}\left (2,\frac {d}{d+\frac {e}{x^{2/3}}}\right )}{2 d^3}+\frac {3 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {d}{d+\frac {e}{x^{2/3}}}\right )}{d^3}+\frac {3 b^3 e^3 n^3 \operatorname {PolyLog}\left (2,1+\frac {e}{d x^{2/3}}\right )}{d^3}+\frac {3 b^3 e^3 n^3 \operatorname {PolyLog}\left (3,\frac {d}{d+\frac {e}{x^{2/3}}}\right )}{d^3} \] Output:

3/2*b^2*e^2*n^2*(d+e/x^(2/3))*x^(2/3)*(a+b*ln(c*(d+e/x^(2/3))^n))/d^3+3/2* 
b^2*e^3*n^2*ln(1-d/(d+e/x^(2/3)))*(a+b*ln(c*(d+e/x^(2/3))^n))/d^3-3/2*b*e^ 
2*n*(d+e/x^(2/3))*x^(2/3)*(a+b*ln(c*(d+e/x^(2/3))^n))^2/d^3+3/4*b*e*n*x^(4 
/3)*(a+b*ln(c*(d+e/x^(2/3))^n))^2/d-3/2*b*e^3*n*ln(1-d/(d+e/x^(2/3)))*(a+b 
*ln(c*(d+e/x^(2/3))^n))^2/d^3+1/2*x^2*(a+b*ln(c*(d+e/x^(2/3))^n))^3+3*b^2* 
e^3*n^2*(a+b*ln(c*(d+e/x^(2/3))^n))*ln(-e/d/x^(2/3))/d^3+b^3*e^3*n^3*ln(x) 
/d^3-3/2*b^3*e^3*n^3*polylog(2,d/(d+e/x^(2/3)))/d^3+3*b^2*e^3*n^2*(a+b*ln( 
c*(d+e/x^(2/3))^n))*polylog(2,d/(d+e/x^(2/3)))/d^3+3*b^3*e^3*n^3*polylog(2 
,1+e/d/x^(2/3))/d^3+3*b^3*e^3*n^3*polylog(3,d/(d+e/x^(2/3)))/d^3
 

Mathematica [F]

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

Integrate[x*(a + b*Log[c*(d + e/x^(2/3))^n])^3,x]
 

Output:

Integrate[x*(a + b*Log[c*(d + e/x^(2/3))^n])^3, x]
 

Rubi [A] (warning: unable to verify)

Time = 3.31 (sec) , antiderivative size = 373, normalized size of antiderivative = 0.83, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {2904, 2845, 2858, 25, 27, 2789, 2756, 2789, 2751, 16, 2755, 2754, 2779, 2821, 2838, 7143}

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.

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

\(\Big \downarrow \) 2904

\(\displaystyle -\frac {3}{2} \int x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3d\frac {1}{x^{2/3}}\)

\(\Big \downarrow \) 2845

\(\displaystyle -\frac {3}{2} \left (b e n \int \frac {x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{d+\frac {e}{x^{2/3}}}d\frac {1}{x^{2/3}}-\frac {1}{3} x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3\right )\)

\(\Big \downarrow \) 2858

\(\displaystyle -\frac {3}{2} \left (b n \int x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2d\left (d+\frac {e}{x^{2/3}}\right )-\frac {1}{3} x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3\right )\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {3}{2} \left (-b n \int -x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2d\left (d+\frac {e}{x^{2/3}}\right )-\frac {1}{3} x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3\right )\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {3}{2} \left (-b e^3 n \int -\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{e^3}d\left (d+\frac {e}{x^{2/3}}\right )-\frac {1}{3} x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3\right )\)

\(\Big \downarrow \) 2789

\(\displaystyle -\frac {3}{2} \left (-b e^3 n \left (\frac {\int -\frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{e^3}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\int \frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{e^2}d\left (d+\frac {e}{x^{2/3}}\right )}{d}\right )-\frac {1}{3} x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3\right )\)

\(\Big \downarrow \) 2756

\(\displaystyle -\frac {3}{2} \left (-b e^3 n \left (\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{2 e^2}-b n \int \frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^2}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\int \frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{e^2}d\left (d+\frac {e}{x^{2/3}}\right )}{d}\right )-\frac {1}{3} x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3\right )\)

\(\Big \downarrow \) 2789

\(\displaystyle -\frac {3}{2} \left (-b e^3 n \left (\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{2 e^2}-b n \left (\frac {\int \frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^2}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e}d\left (d+\frac {e}{x^{2/3}}\right )}{d}\right )}{d}+\frac {\frac {\int \frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{e^2}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{e}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}\right )-\frac {1}{3} x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3\right )\)

\(\Big \downarrow \) 2751

\(\displaystyle -\frac {3}{2} \left (-b e^3 n \left (\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{2 e^2}-b n \left (\frac {-\frac {b n \int -\frac {x^{2/3}}{e}d\left (d+\frac {e}{x^{2/3}}\right )}{d}-\frac {x^{2/3} \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d e}}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e}d\left (d+\frac {e}{x^{2/3}}\right )}{d}\right )}{d}+\frac {\frac {\int \frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{e^2}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{e}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}\right )-\frac {1}{3} x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3\right )\)

\(\Big \downarrow \) 16

\(\displaystyle -\frac {3}{2} \left (-b e^3 n \left (\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{2 e^2}-b n \left (\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\frac {b n \log \left (-\frac {e}{x^{2/3}}\right )}{d}-\frac {x^{2/3} \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d e}}{d}\right )}{d}+\frac {\frac {\int \frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{e^2}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{e}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}\right )-\frac {1}{3} x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3\right )\)

\(\Big \downarrow \) 2755

\(\displaystyle -\frac {3}{2} \left (-b e^3 n \left (\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{2 e^2}-b n \left (\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\frac {b n \log \left (-\frac {e}{x^{2/3}}\right )}{d}-\frac {x^{2/3} \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d e}}{d}\right )}{d}+\frac {\frac {-\frac {2 b n \int -\frac {x^{2/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e}d\left (d+\frac {e}{x^{2/3}}\right )}{d}-\frac {x^{2/3} \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{d e}}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{e}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}\right )-\frac {1}{3} x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3\right )\)

\(\Big \downarrow \) 2754

\(\displaystyle -\frac {3}{2} \left (-b e^3 n \left (\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{2 e^2}-b n \left (\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\frac {b n \log \left (-\frac {e}{x^{2/3}}\right )}{d}-\frac {x^{2/3} \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d e}}{d}\right )}{d}+\frac {\frac {-\frac {2 b n \left (b n \int x^{2/3} \log \left (1-\frac {d+\frac {e}{x^{2/3}}}{d}\right )d\left (d+\frac {e}{x^{2/3}}\right )-\log \left (1-\frac {d+\frac {e}{x^{2/3}}}{d}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )\right )}{d}-\frac {x^{2/3} \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{d e}}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{e}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}\right )-\frac {1}{3} x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3\right )\)

\(\Big \downarrow \) 2779

\(\displaystyle -\frac {3}{2} \left (-b e^3 n \left (\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{2 e^2}-b n \left (\frac {\frac {b n \int x^{2/3} \log \left (1-d x^{2/3}\right )d\left (d+\frac {e}{x^{2/3}}\right )}{d}-\frac {\log \left (1-d x^{2/3}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d}}{d}+\frac {\frac {b n \log \left (-\frac {e}{x^{2/3}}\right )}{d}-\frac {x^{2/3} \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d e}}{d}\right )}{d}+\frac {\frac {-\frac {2 b n \left (b n \int x^{2/3} \log \left (1-\frac {d+\frac {e}{x^{2/3}}}{d}\right )d\left (d+\frac {e}{x^{2/3}}\right )-\log \left (1-\frac {d+\frac {e}{x^{2/3}}}{d}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )\right )}{d}-\frac {x^{2/3} \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{d e}}{d}+\frac {\frac {2 b n \int x^{2/3} \log \left (1-d x^{2/3}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )d\left (d+\frac {e}{x^{2/3}}\right )}{d}-\frac {\log \left (1-d x^{2/3}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{d}}{d}}{d}\right )-\frac {1}{3} x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3\right )\)

\(\Big \downarrow \) 2821

\(\displaystyle -\frac {3}{2} \left (-b e^3 n \left (\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{2 e^2}-b n \left (\frac {\frac {b n \int x^{2/3} \log \left (1-d x^{2/3}\right )d\left (d+\frac {e}{x^{2/3}}\right )}{d}-\frac {\log \left (1-d x^{2/3}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d}}{d}+\frac {\frac {b n \log \left (-\frac {e}{x^{2/3}}\right )}{d}-\frac {x^{2/3} \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d e}}{d}\right )}{d}+\frac {\frac {\frac {2 b n \left (\operatorname {PolyLog}\left (2,d x^{2/3}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )-b n \int x^{2/3} \operatorname {PolyLog}\left (2,d x^{2/3}\right )d\left (d+\frac {e}{x^{2/3}}\right )\right )}{d}-\frac {\log \left (1-d x^{2/3}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{d}}{d}+\frac {-\frac {2 b n \left (b n \int x^{2/3} \log \left (1-\frac {d+\frac {e}{x^{2/3}}}{d}\right )d\left (d+\frac {e}{x^{2/3}}\right )-\log \left (1-\frac {d+\frac {e}{x^{2/3}}}{d}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )\right )}{d}-\frac {x^{2/3} \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{d e}}{d}}{d}\right )-\frac {1}{3} x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3\right )\)

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {3}{2} \left (-b e^3 n \left (\frac {\frac {\frac {2 b n \left (\operatorname {PolyLog}\left (2,d x^{2/3}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )-b n \int x^{2/3} \operatorname {PolyLog}\left (2,d x^{2/3}\right )d\left (d+\frac {e}{x^{2/3}}\right )\right )}{d}-\frac {\log \left (1-d x^{2/3}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{d}}{d}+\frac {-\frac {2 b n \left (-\log \left (1-\frac {d+\frac {e}{x^{2/3}}}{d}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )-b n \operatorname {PolyLog}\left (2,\frac {d+\frac {e}{x^{2/3}}}{d}\right )\right )}{d}-\frac {x^{2/3} \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{d e}}{d}}{d}+\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{2 e^2}-b n \left (\frac {\frac {b n \log \left (-\frac {e}{x^{2/3}}\right )}{d}-\frac {x^{2/3} \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d e}}{d}+\frac {\frac {b n \operatorname {PolyLog}\left (2,d x^{2/3}\right )}{d}-\frac {\log \left (1-d x^{2/3}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d}}{d}\right )}{d}\right )-\frac {1}{3} x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3\right )\)

\(\Big \downarrow \) 7143

\(\displaystyle -\frac {3}{2} \left (-b e^3 n \left (\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{2 e^2}-b n \left (\frac {\frac {b n \log \left (-\frac {e}{x^{2/3}}\right )}{d}-\frac {x^{2/3} \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d e}}{d}+\frac {\frac {b n \operatorname {PolyLog}\left (2,d x^{2/3}\right )}{d}-\frac {\log \left (1-d x^{2/3}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d}}{d}\right )}{d}+\frac {\frac {-\frac {2 b n \left (-\log \left (1-\frac {d+\frac {e}{x^{2/3}}}{d}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )-b n \operatorname {PolyLog}\left (2,\frac {d+\frac {e}{x^{2/3}}}{d}\right )\right )}{d}-\frac {x^{2/3} \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{d e}}{d}+\frac {\frac {2 b n \left (\operatorname {PolyLog}\left (2,d x^{2/3}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )+b n \operatorname {PolyLog}\left (3,d x^{2/3}\right )\right )}{d}-\frac {\log \left (1-d x^{2/3}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{d}}{d}}{d}\right )-\frac {1}{3} x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3\right )\)

Input:

Int[x*(a + b*Log[c*(d + e/x^(2/3))^n])^3,x]
 

Output:

(-3*(-1/3*(x^2*(a + b*Log[c*(d + e/x^(2/3))^n])^3) - b*e^3*n*(((x^(4/3)*(a 
 + b*Log[c/x^((2*n)/3)])^2)/(2*e^2) - b*n*(((b*n*Log[-(e/x^(2/3))])/d - (( 
d + e/x^(2/3))*x^(2/3)*(a + b*Log[c/x^((2*n)/3)]))/(d*e))/d + (-((Log[1 - 
d*x^(2/3)]*(a + b*Log[c/x^((2*n)/3)]))/d) + (b*n*PolyLog[2, d*x^(2/3)])/d) 
/d))/d + ((-(((d + e/x^(2/3))*x^(2/3)*(a + b*Log[c/x^((2*n)/3)])^2)/(d*e)) 
 - (2*b*n*(-(Log[1 - (d + e/x^(2/3))/d]*(a + b*Log[c/x^((2*n)/3)])) - b*n* 
PolyLog[2, (d + e/x^(2/3))/d]))/d)/d + (-((Log[1 - d*x^(2/3)]*(a + b*Log[c 
/x^((2*n)/3)])^2)/d) + (2*b*n*((a + b*Log[c/x^((2*n)/3)])*PolyLog[2, d*x^( 
2/3)] + b*n*PolyLog[3, d*x^(2/3)]))/d)/d)/d)))/2
 

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 2751

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x 
_Symbol] :> Simp[x*(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])/d), x] - Simp[b* 
(n/d)   Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q, r}, 
x] && EqQ[r*(q + 1) + 1, 0]
 

rule 2754

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symb 
ol] :> Simp[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^p/e), x] - Simp[b*n*(p/e) 
  Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, 
b, c, d, e, n}, x] && IGtQ[p, 0]
 

rule 2755

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_))^2, x_Sy 
mbol] :> Simp[x*((a + b*Log[c*x^n])^p/(d*(d + e*x))), x] - Simp[b*n*(p/d) 
 Int[(a + b*Log[c*x^n])^(p - 1)/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, 
n, p}, x] && GtQ[p, 0]
 

rule 2756

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), 
x_Symbol] :> Simp[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] 
- Simp[b*n*(p/(e*(q + 1)))   Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 
 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, 
 -1] && (EqQ[p, 1] || (IntegersQ[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] & 
& NeQ[q, 1]))
 

rule 2779

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r 
_.))), x_Symbol] :> Simp[(-Log[1 + d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)) 
, x] + Simp[b*n*(p/(d*r))   Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^(p - 
 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]
 

rule 2789

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/ 
(x_), x_Symbol] :> Simp[1/d   Int[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x 
), x], x] - Simp[e/d   Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; Free 
Q[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]
 

rule 2821

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

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 2845

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

rule 2858

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_ 
.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))^(r_.), x_Symbol] :> Simp[1/e   Subst[In 
t[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x, d + 
e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - 
d*g, 0] && (IGtQ[p, 0] || IGtQ[r, 0]) && IntegerQ[2*r]
 

rule 2904

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m 
_.), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*L 
og[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, 
 x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) & 
&  !(EqQ[q, 1] && ILtQ[n, 0] && IGtQ[m, 0])
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

Maple [F]

\[\int x {\left (a +b \ln \left (c \left (d +\frac {e}{x^{\frac {2}{3}}}\right )^{n}\right )\right )}^{3}d x\]

Input:

int(x*(a+b*ln(c*(d+e/x^(2/3))^n))^3,x)
 

Output:

int(x*(a+b*ln(c*(d+e/x^(2/3))^n))^3,x)
 

Fricas [F]

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

integrate(x*(a+b*log(c*(d+e/x^(2/3))^n))^3,x, algorithm="fricas")
 

Output:

integral(b^3*x*log(c*((d*x + e*x^(1/3))/x)^n)^3 + 3*a*b^2*x*log(c*((d*x + 
e*x^(1/3))/x)^n)^2 + 3*a^2*b*x*log(c*((d*x + e*x^(1/3))/x)^n) + a^3*x, x)
 

Sympy [F(-1)]

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

integrate(x*(a+b*ln(c*(d+e/x**(2/3))**n))**3,x)
 

Output:

Timed out
 

Maxima [F]

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

integrate(x*(a+b*log(c*(d+e/x^(2/3))^n))^3,x, algorithm="maxima")
 

Output:

1/2*b^3*x^2*log((d*x^(2/3) + e)^n)^3 - integrate((8*(b^3*d*x^2 + b^3*e*x^( 
4/3))*log(x^(1/3*n))^3 - (b^3*d*log(c)^3 + 3*a*b^2*d*log(c)^2 + 3*a^2*b*d* 
log(c) + a^3*d)*x^2 + (b^3*d*n*x^2 - 3*(b^3*d*log(c) + a*b^2*d)*x^2 - 3*(b 
^3*e*log(c) + a*b^2*e)*x^(4/3) + 6*(b^3*d*x^2 + b^3*e*x^(4/3))*log(x^(1/3* 
n)))*log((d*x^(2/3) + e)^n)^2 - 12*((b^3*d*log(c) + a*b^2*d)*x^2 + (b^3*e* 
log(c) + a*b^2*e)*x^(4/3))*log(x^(1/3*n))^2 - (b^3*e*log(c)^3 + 3*a*b^2*e* 
log(c)^2 + 3*a^2*b*e*log(c) + a^3*e)*x^(4/3) - 3*((b^3*d*log(c)^2 + 2*a*b^ 
2*d*log(c) + a^2*b*d)*x^2 + 4*(b^3*d*x^2 + b^3*e*x^(4/3))*log(x^(1/3*n))^2 
 + (b^3*e*log(c)^2 + 2*a*b^2*e*log(c) + a^2*b*e)*x^(4/3) - 4*((b^3*d*log(c 
) + a*b^2*d)*x^2 + (b^3*e*log(c) + a*b^2*e)*x^(4/3))*log(x^(1/3*n)))*log(( 
d*x^(2/3) + e)^n) + 6*((b^3*d*log(c)^2 + 2*a*b^2*d*log(c) + a^2*b*d)*x^2 + 
 (b^3*e*log(c)^2 + 2*a*b^2*e*log(c) + a^2*b*e)*x^(4/3))*log(x^(1/3*n)))/(d 
*x + e*x^(1/3)), x)
 

Giac [F]

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

integrate(x*(a+b*log(c*(d+e/x^(2/3))^n))^3,x, algorithm="giac")
 

Output:

integrate((b*log(c*(d + e/x^(2/3))^n) + a)^3*x, x)
 

Mupad [F(-1)]

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

int(x*(a + b*log(c*(d + e/x^(2/3))^n))^3,x)
 

Output:

int(x*(a + b*log(c*(d + e/x^(2/3))^n))^3, x)
 

Reduce [F]

\[ \int x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx=\frac {-6 x^{\frac {2}{3}} a^{2} b d \,e^{2} n +6 x^{\frac {2}{3}} a \,b^{2} d \,e^{2} n^{2}+3 x^{\frac {4}{3}} {\mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right )}^{2} b^{3} d^{2} e n +6 x^{\frac {4}{3}} \mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right ) a \,b^{2} d^{2} e n +3 x^{\frac {4}{3}} a^{2} b \,d^{2} e n -4 \left (\int \frac {x^{\frac {1}{3}} {\mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right )}^{2}}{x^{\frac {2}{3}} d +e}d x \right ) b^{3} d^{2} e^{2} n -8 \left (\int \frac {x^{\frac {1}{3}} \mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right )}{x^{\frac {2}{3}} d +e}d x \right ) a \,b^{2} d^{2} e^{2} n +4 \left (\int \frac {x^{\frac {1}{3}} \mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right )}{x^{\frac {2}{3}} d +e}d x \right ) b^{3} d^{2} e^{2} n^{2}+12 \,\mathrm {log}\left (x^{\frac {1}{3}}\right ) a^{2} b \,e^{3} n -12 \,\mathrm {log}\left (x^{\frac {1}{3}}\right ) a \,b^{2} e^{3} n^{2}+2 {\mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right )}^{3} b^{3} d^{3} x^{2}+6 {\mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right )}^{2} a \,b^{2} d^{3} x^{2}+6 \,\mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right ) a^{2} b \,d^{3} x^{2}+6 \,\mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right ) a^{2} b \,e^{3}-6 \,\mathrm {log}\left (\frac {\left (x^{\frac {2}{3}} d +e \right )^{n} c}{x^{\frac {2 n}{3}}}\right ) a \,b^{2} e^{3} n +2 a^{3} d^{3} x^{2}}{4 d^{3}} \] Input:

int(x*(a+b*log(c*(d+e/x^(2/3))^n))^3,x)
                                                                                    
                                                                                    
 

Output:

( - 6*x**(2/3)*a**2*b*d*e**2*n + 6*x**(2/3)*a*b**2*d*e**2*n**2 + 3*x**(1/3 
)*log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))**2*b**3*d**2*e*n*x + 6*x**(1/3 
)*log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))*a*b**2*d**2*e*n*x + 3*x**(1/3) 
*a**2*b*d**2*e*n*x - 4*int((x**(1/3)*log(((x**(2/3)*d + e)**n*c)/x**((2*n) 
/3))**2)/(x**(2/3)*d + e),x)*b**3*d**2*e**2*n - 8*int((x**(1/3)*log(((x**( 
2/3)*d + e)**n*c)/x**((2*n)/3)))/(x**(2/3)*d + e),x)*a*b**2*d**2*e**2*n + 
4*int((x**(1/3)*log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3)))/(x**(2/3)*d + e 
),x)*b**3*d**2*e**2*n**2 + 12*log(x**(1/3))*a**2*b*e**3*n - 12*log(x**(1/3 
))*a*b**2*e**3*n**2 + 2*log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))**3*b**3* 
d**3*x**2 + 6*log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))**2*a*b**2*d**3*x** 
2 + 6*log(((x**(2/3)*d + e)**n*c)/x**((2*n)/3))*a**2*b*d**3*x**2 + 6*log(( 
(x**(2/3)*d + e)**n*c)/x**((2*n)/3))*a**2*b*e**3 - 6*log(((x**(2/3)*d + e) 
**n*c)/x**((2*n)/3))*a*b**2*e**3*n + 2*a**3*d**3*x**2)/(4*d**3)