3.5.61 \(\int \frac {(a+b \log (c (d+e \sqrt [3]{x})^n))^3}{x^2} \, dx\) [461]

3.5.61.1 Optimal result

Integrand size = 24, antiderivative size = 439 \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x^2} \, dx=-\frac {3 b^2 e^2 n^2 \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{d^3 \sqrt [3]{x}}-\frac {3 b^2 e^3 n^2 \log \left (1-\frac {d}{d+e \sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{d^3}-\frac {3 b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d x^{2/3}}+\frac {3 b e^2 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{d^3 \sqrt [3]{x}}+\frac {3 b e^3 n \log \left (1-\frac {d}{d+e \sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{d^3}-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x}-\frac {6 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \log \left (-\frac {e \sqrt [3]{x}}{d}\right )}{d^3}+\frac {b^3 e^3 n^3 \log (x)}{d^3}+\frac {3 b^3 e^3 n^3 \operatorname {PolyLog}\left (2,\frac {d}{d+e \sqrt [3]{x}}\right )}{d^3}-\frac {6 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {d}{d+e \sqrt [3]{x}}\right )}{d^3}-\frac {6 b^3 e^3 n^3 \operatorname {PolyLog}\left (2,1+\frac {e \sqrt [3]{x}}{d}\right )}{d^3}-\frac {6 b^3 e^3 n^3 \operatorname {PolyLog}\left (3,\frac {d}{d+e \sqrt [3]{x}}\right )}{d^3} \]

output

-3*b^2*e^2*n^2*(d+e*x^(1/3))*(a+b*ln(c*(d+e*x^(1/3))^n))/d^3/x^(1/3)-3*b^2 
*e^3*n^2*ln(1-d/(d+e*x^(1/3)))*(a+b*ln(c*(d+e*x^(1/3))^n))/d^3-3/2*b*e*n*( 
a+b*ln(c*(d+e*x^(1/3))^n))^2/d/x^(2/3)+3*b*e^2*n*(d+e*x^(1/3))*(a+b*ln(c*( 
d+e*x^(1/3))^n))^2/d^3/x^(1/3)+3*b*e^3*n*ln(1-d/(d+e*x^(1/3)))*(a+b*ln(c*( 
d+e*x^(1/3))^n))^2/d^3-(a+b*ln(c*(d+e*x^(1/3))^n))^3/x-6*b^2*e^3*n^2*(a+b* 
ln(c*(d+e*x^(1/3))^n))*ln(-e*x^(1/3)/d)/d^3+b^3*e^3*n^3*ln(x)/d^3+3*b^3*e^ 
3*n^3*polylog(2,d/(d+e*x^(1/3)))/d^3-6*b^2*e^3*n^2*(a+b*ln(c*(d+e*x^(1/3)) 
^n))*polylog(2,d/(d+e*x^(1/3)))/d^3-6*b^3*e^3*n^3*polylog(2,1+e*x^(1/3)/d) 
/d^3-6*b^3*e^3*n^3*polylog(3,d/(d+e*x^(1/3)))/d^3
 

3.5.61.2 Mathematica [A] (verified)

Time = 0.61 (sec) , antiderivative size = 733, normalized size of antiderivative = 1.67 \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x^2} \, dx=\frac {-3 b d^2 e n \sqrt [3]{x} \left (a-b n \log \left (d+e \sqrt [3]{x}\right )+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2+6 b d e^2 n x^{2/3} \left (a-b n \log \left (d+e \sqrt [3]{x}\right )+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2-6 b d^3 n \log \left (d+e \sqrt [3]{x}\right ) \left (a-b n \log \left (d+e \sqrt [3]{x}\right )+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2-6 b e^3 n x \log \left (d+e \sqrt [3]{x}\right ) \left (a-b n \log \left (d+e \sqrt [3]{x}\right )+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2-2 d^3 \left (a-b n \log \left (d+e \sqrt [3]{x}\right )+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3+2 b e^3 n x \left (a-b n \log \left (d+e \sqrt [3]{x}\right )+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \log (x)-6 b^2 n^2 \left (a-b n \log \left (d+e \sqrt [3]{x}\right )+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \left (d e^2 x^{2/3}+\left (d^3+e^3 x\right ) \log ^2\left (d+e \sqrt [3]{x}\right )+3 e^3 x \log \left (-\frac {e \sqrt [3]{x}}{d}\right )+\log \left (d+e \sqrt [3]{x}\right ) \left (d^2 e \sqrt [3]{x}-2 d e^2 x^{2/3}-3 e^3 x-2 e^3 x \log \left (-\frac {e \sqrt [3]{x}}{d}\right )\right )-2 e^3 x \operatorname {PolyLog}\left (2,1+\frac {e \sqrt [3]{x}}{d}\right )\right )+b^3 n^3 \left (-6 d e^2 x^{2/3} \log \left (d+e \sqrt [3]{x}\right )-6 e^3 x \log \left (d+e \sqrt [3]{x}\right )-3 d^2 e \sqrt [3]{x} \log ^2\left (d+e \sqrt [3]{x}\right )+6 d e^2 x^{2/3} \log ^2\left (d+e \sqrt [3]{x}\right )+9 e^3 x \log ^2\left (d+e \sqrt [3]{x}\right )-2 d^3 \log ^3\left (d+e \sqrt [3]{x}\right )-2 e^3 x \log ^3\left (d+e \sqrt [3]{x}\right )+6 e^3 x \log \left (-\frac {e \sqrt [3]{x}}{d}\right )-18 e^3 x \log \left (d+e \sqrt [3]{x}\right ) \log \left (-\frac {e \sqrt [3]{x}}{d}\right )+6 e^3 x \log ^2\left (d+e \sqrt [3]{x}\right ) \log \left (-\frac {e \sqrt [3]{x}}{d}\right )+6 e^3 x \left (-3+2 \log \left (d+e \sqrt [3]{x}\right )\right ) \operatorname {PolyLog}\left (2,1+\frac {e \sqrt [3]{x}}{d}\right )-12 e^3 x \operatorname {PolyLog}\left (3,1+\frac {e \sqrt [3]{x}}{d}\right )\right )}{2 d^3 x} \]

input

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

output

(-3*b*d^2*e*n*x^(1/3)*(a - b*n*Log[d + e*x^(1/3)] + b*Log[c*(d + e*x^(1/3) 
)^n])^2 + 6*b*d*e^2*n*x^(2/3)*(a - b*n*Log[d + e*x^(1/3)] + b*Log[c*(d + e 
*x^(1/3))^n])^2 - 6*b*d^3*n*Log[d + e*x^(1/3)]*(a - b*n*Log[d + e*x^(1/3)] 
 + b*Log[c*(d + e*x^(1/3))^n])^2 - 6*b*e^3*n*x*Log[d + e*x^(1/3)]*(a - b*n 
*Log[d + e*x^(1/3)] + b*Log[c*(d + e*x^(1/3))^n])^2 - 2*d^3*(a - b*n*Log[d 
 + e*x^(1/3)] + b*Log[c*(d + e*x^(1/3))^n])^3 + 2*b*e^3*n*x*(a - b*n*Log[d 
 + e*x^(1/3)] + b*Log[c*(d + e*x^(1/3))^n])^2*Log[x] - 6*b^2*n^2*(a - b*n* 
Log[d + e*x^(1/3)] + b*Log[c*(d + e*x^(1/3))^n])*(d*e^2*x^(2/3) + (d^3 + e 
^3*x)*Log[d + e*x^(1/3)]^2 + 3*e^3*x*Log[-((e*x^(1/3))/d)] + Log[d + e*x^( 
1/3)]*(d^2*e*x^(1/3) - 2*d*e^2*x^(2/3) - 3*e^3*x - 2*e^3*x*Log[-((e*x^(1/3 
))/d)]) - 2*e^3*x*PolyLog[2, 1 + (e*x^(1/3))/d]) + b^3*n^3*(-6*d*e^2*x^(2/ 
3)*Log[d + e*x^(1/3)] - 6*e^3*x*Log[d + e*x^(1/3)] - 3*d^2*e*x^(1/3)*Log[d 
 + e*x^(1/3)]^2 + 6*d*e^2*x^(2/3)*Log[d + e*x^(1/3)]^2 + 9*e^3*x*Log[d + e 
*x^(1/3)]^2 - 2*d^3*Log[d + e*x^(1/3)]^3 - 2*e^3*x*Log[d + e*x^(1/3)]^3 + 
6*e^3*x*Log[-((e*x^(1/3))/d)] - 18*e^3*x*Log[d + e*x^(1/3)]*Log[-((e*x^(1/ 
3))/d)] + 6*e^3*x*Log[d + e*x^(1/3)]^2*Log[-((e*x^(1/3))/d)] + 6*e^3*x*(-3 
 + 2*Log[d + e*x^(1/3)])*PolyLog[2, 1 + (e*x^(1/3))/d] - 12*e^3*x*PolyLog[ 
3, 1 + (e*x^(1/3))/d]))/(2*d^3*x)
 

3.5.61.3 Rubi [A] (warning: unable to verify)

Time = 1.87 (sec) , antiderivative size = 371, normalized size of antiderivative = 0.85, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, 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.

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

\(\Big \downarrow \) 2904

\(\displaystyle 3 \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x^{4/3}}d\sqrt [3]{x}\)

\(\Big \downarrow \) 2845

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

\(\Big \downarrow \) 2858

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2789

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

\(\Big \downarrow \) 2756

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

\(\Big \downarrow \) 2789

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

\(\Big \downarrow \) 2751

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 2755

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

\(\Big \downarrow \) 2754

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

\(\Big \downarrow \) 2779

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

\(\Big \downarrow \) 2821

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 7143

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

input

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

output

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

3.5.61.3.1 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]
 

3.5.61.4 Maple [F]

\[\int \frac {{\left (a +b \ln \left (c \left (d +e \,x^{\frac {1}{3}}\right )^{n}\right )\right )}^{3}}{x^{2}}d x\]

input

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

output

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

3.5.61.5 Fricas [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right ) + a\right )}^{3}}{x^{2}} \,d x } \]

input

integrate((a+b*log(c*(d+e*x^(1/3))^n))^3/x^2,x, algorithm="fricas")
 

output

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

3.5.61.6 Sympy [F]

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

input

integrate((a+b*ln(c*(d+e*x**(1/3))**n))**3/x**2,x)
 

output

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

3.5.61.7 Maxima [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right ) + a\right )}^{3}}{x^{2}} \,d x } \]

input

integrate((a+b*log(c*(d+e*x^(1/3))^n))^3/x^2,x, algorithm="maxima")
 

output

-1/2*(2*b^3*d^3*x^(2/3)*log((e*x^(1/3) + d)^n)^3 + (6*b^3*e^3*n*x^(5/3)*lo 
g(e*x^(1/3) + d) - 6*b^3*d*e^2*n*x^(4/3) + 3*b^3*d^2*e*n*x - 2*(b^3*e^3*n* 
x*log(x) - 3*b^3*d^3*log(c) - 3*a*b^2*d^3)*x^(2/3))*log((e*x^(1/3) + d)^n) 
^2)/(d^3*x^(5/3)) + integrate(1/3*(3*(b^3*d^3*e*log(c)^3 + 3*a*b^2*d^3*e*l 
og(c)^2 + 3*a^2*b*d^3*e*log(c) + a^3*d^3*e)*x^(5/3) + 3*(b^3*d^4*log(c)^3 
+ 3*a*b^2*d^4*log(c)^2 + 3*a^2*b*d^4*log(c) + a^3*d^4)*x^(4/3) + (6*b^3*e^ 
4*n^2*x^(8/3)*log(e*x^(1/3) + d) - 6*b^3*d*e^3*n^2*x^(7/3) + 3*b^3*d^2*e^2 
*n^2*x^2 + 9*(b^3*d^3*e*log(c)^2 + 2*a*b^2*d^3*e*log(c) + a^2*b*d^3*e)*x^( 
5/3) + 9*(b^3*d^4*log(c)^2 + 2*a*b^2*d^4*log(c) + a^2*b*d^4)*x^(4/3) - 2*( 
b^3*e^4*n^2*x^2*log(x) - 3*(b^3*d^3*e*n*log(c) + a*b^2*d^3*e*n)*x)*x^(2/3) 
)*log((e*x^(1/3) + d)^n))/(d^3*e*x^(11/3) + d^4*x^(10/3)), x)
 

3.5.61.8 Giac [F]

\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right ) + a\right )}^{3}}{x^{2}} \,d x } \]

input

integrate((a+b*log(c*(d+e*x^(1/3))^n))^3/x^2,x, algorithm="giac")
 

output

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

3.5.61.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x^2} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )\right )}^3}{x^2} \,d x \]

input

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

output

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