\(\int x^2 (a+b \log (c x^n)) \operatorname {PolyLog}(3,e x) \, dx\) [220]

Optimal result

Integrand size = 19, antiderivative size = 253 \[ \int x^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(3,e x) \, dx=-\frac {2 b n x}{27 e^2}-\frac {b n x^2}{36 e}-\frac {4}{243} b n x^3+\frac {x \left (a+b \log \left (c x^n\right )\right )}{27 e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{54 e}+\frac {1}{81} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {b n \log (1-e x)}{27 e^3}+\frac {1}{27} b n x^3 \log (1-e x)+\frac {\left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{27 e^3}-\frac {1}{27} x^3 \left (a+b \log \left (c x^n\right )\right ) \log (1-e x)+\frac {b n \operatorname {PolyLog}(2,e x)}{27 e^3}+\frac {2}{27} b n x^3 \operatorname {PolyLog}(2,e x)-\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(2,e x)-\frac {1}{9} b n x^3 \operatorname {PolyLog}(3,e x)+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(3,e x) \] Output:

-2/27*b*n*x/e^2-1/36*b*n*x^2/e-4/243*b*n*x^3+1/27*x*(a+b*ln(c*x^n))/e^2+1/ 
54*x^2*(a+b*ln(c*x^n))/e+1/81*x^3*(a+b*ln(c*x^n))-1/27*b*n*ln(-e*x+1)/e^3+ 
1/27*b*n*x^3*ln(-e*x+1)+1/27*(a+b*ln(c*x^n))*ln(-e*x+1)/e^3-1/27*x^3*(a+b* 
ln(c*x^n))*ln(-e*x+1)+1/27*b*n*polylog(2,e*x)/e^3+2/27*b*n*x^3*polylog(2,e 
*x)-1/9*x^3*(a+b*ln(c*x^n))*polylog(2,e*x)-1/9*b*n*x^3*polylog(3,e*x)+1/3* 
x^3*(a+b*ln(c*x^n))*polylog(3,e*x)
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 1.18 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.59, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.684, Rules used = {2832, 2832, 25, 2823, 2009, 2842, 49, 2009, 7145, 25, 2842, 49, 2009}

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.

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

\(\Big \downarrow \) 2832

\(\displaystyle -\frac {1}{3} \int x^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(2,e x)dx+\frac {1}{9} b n \int x^2 \operatorname {PolyLog}(2,e x)dx+\frac {1}{3} x^3 \operatorname {PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{9} b n x^3 \operatorname {PolyLog}(3,e x)\)

\(\Big \downarrow \) 2832

\(\displaystyle \frac {1}{3} \left (\frac {1}{3} \int -x^2 \left (a+b \log \left (c x^n\right )\right ) \log (1-e x)dx-\frac {1}{9} b n \int -x^2 \log (1-e x)dx-\frac {1}{3} x^3 \operatorname {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )+\frac {1}{9} b n x^3 \operatorname {PolyLog}(2,e x)\right )+\frac {1}{9} b n \int x^2 \operatorname {PolyLog}(2,e x)dx+\frac {1}{3} x^3 \operatorname {PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{9} b n x^3 \operatorname {PolyLog}(3,e x)\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{3} \left (-\frac {1}{3} \int x^2 \left (a+b \log \left (c x^n\right )\right ) \log (1-e x)dx+\frac {1}{9} b n \int x^2 \log (1-e x)dx-\frac {1}{3} x^3 \operatorname {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )+\frac {1}{9} b n x^3 \operatorname {PolyLog}(2,e x)\right )+\frac {1}{9} b n \int x^2 \operatorname {PolyLog}(2,e x)dx+\frac {1}{3} x^3 \operatorname {PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{9} b n x^3 \operatorname {PolyLog}(3,e x)\)

\(\Big \downarrow \) 2823

\(\displaystyle \frac {1}{3} \left (\frac {1}{3} \left (b n \int \left (\frac {1}{3} \log (1-e x) x^2-\frac {x^2}{9}-\frac {x}{6 e}-\frac {1}{3 e^2}-\frac {\log (1-e x)}{3 e^3 x}\right )dx+\frac {\log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{3 e^2}-\frac {1}{3} x^3 \log (1-e x) \left (a+b \log \left (c x^n\right )\right )+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{6 e}+\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )\right )+\frac {1}{9} b n \int x^2 \log (1-e x)dx-\frac {1}{3} x^3 \operatorname {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )+\frac {1}{9} b n x^3 \operatorname {PolyLog}(2,e x)\right )+\frac {1}{9} b n \int x^2 \operatorname {PolyLog}(2,e x)dx+\frac {1}{3} x^3 \operatorname {PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{9} b n x^3 \operatorname {PolyLog}(3,e x)\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{3} \left (\frac {1}{9} b n \int x^2 \log (1-e x)dx+\frac {1}{3} \left (\frac {\log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{3 e^2}-\frac {1}{3} x^3 \log (1-e x) \left (a+b \log \left (c x^n\right )\right )+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{6 e}+\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )+b n \left (\frac {\operatorname {PolyLog}(2,e x)}{3 e^3}-\frac {\log (1-e x)}{9 e^3}-\frac {4 x}{9 e^2}+\frac {1}{9} x^3 \log (1-e x)-\frac {5 x^2}{36 e}-\frac {2 x^3}{27}\right )\right )-\frac {1}{3} x^3 \operatorname {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )+\frac {1}{9} b n x^3 \operatorname {PolyLog}(2,e x)\right )+\frac {1}{9} b n \int x^2 \operatorname {PolyLog}(2,e x)dx+\frac {1}{3} x^3 \operatorname {PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{9} b n x^3 \operatorname {PolyLog}(3,e x)\)

\(\Big \downarrow \) 2842

\(\displaystyle \frac {1}{3} \left (\frac {1}{9} b n \left (\frac {1}{3} e \int \frac {x^3}{1-e x}dx+\frac {1}{3} x^3 \log (1-e x)\right )+\frac {1}{3} \left (\frac {\log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{3 e^2}-\frac {1}{3} x^3 \log (1-e x) \left (a+b \log \left (c x^n\right )\right )+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{6 e}+\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )+b n \left (\frac {\operatorname {PolyLog}(2,e x)}{3 e^3}-\frac {\log (1-e x)}{9 e^3}-\frac {4 x}{9 e^2}+\frac {1}{9} x^3 \log (1-e x)-\frac {5 x^2}{36 e}-\frac {2 x^3}{27}\right )\right )-\frac {1}{3} x^3 \operatorname {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )+\frac {1}{9} b n x^3 \operatorname {PolyLog}(2,e x)\right )+\frac {1}{9} b n \int x^2 \operatorname {PolyLog}(2,e x)dx+\frac {1}{3} x^3 \operatorname {PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{9} b n x^3 \operatorname {PolyLog}(3,e x)\)

\(\Big \downarrow \) 49

\(\displaystyle \frac {1}{3} \left (\frac {1}{9} b n \left (\frac {1}{3} e \int \left (-\frac {x^2}{e}-\frac {x}{e^2}-\frac {1}{e^3 (e x-1)}-\frac {1}{e^3}\right )dx+\frac {1}{3} x^3 \log (1-e x)\right )+\frac {1}{3} \left (\frac {\log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{3 e^2}-\frac {1}{3} x^3 \log (1-e x) \left (a+b \log \left (c x^n\right )\right )+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{6 e}+\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )+b n \left (\frac {\operatorname {PolyLog}(2,e x)}{3 e^3}-\frac {\log (1-e x)}{9 e^3}-\frac {4 x}{9 e^2}+\frac {1}{9} x^3 \log (1-e x)-\frac {5 x^2}{36 e}-\frac {2 x^3}{27}\right )\right )-\frac {1}{3} x^3 \operatorname {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )+\frac {1}{9} b n x^3 \operatorname {PolyLog}(2,e x)\right )+\frac {1}{9} b n \int x^2 \operatorname {PolyLog}(2,e x)dx+\frac {1}{3} x^3 \operatorname {PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{9} b n x^3 \operatorname {PolyLog}(3,e x)\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{9} b n \int x^2 \operatorname {PolyLog}(2,e x)dx+\frac {1}{3} \left (\frac {1}{3} \left (\frac {\log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{3 e^2}-\frac {1}{3} x^3 \log (1-e x) \left (a+b \log \left (c x^n\right )\right )+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{6 e}+\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )+b n \left (\frac {\operatorname {PolyLog}(2,e x)}{3 e^3}-\frac {\log (1-e x)}{9 e^3}-\frac {4 x}{9 e^2}+\frac {1}{9} x^3 \log (1-e x)-\frac {5 x^2}{36 e}-\frac {2 x^3}{27}\right )\right )-\frac {1}{3} x^3 \operatorname {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )+\frac {1}{9} b n \left (\frac {1}{3} e \left (-\frac {\log (1-e x)}{e^4}-\frac {x}{e^3}-\frac {x^2}{2 e^2}-\frac {x^3}{3 e}\right )+\frac {1}{3} x^3 \log (1-e x)\right )+\frac {1}{9} b n x^3 \operatorname {PolyLog}(2,e x)\right )+\frac {1}{3} x^3 \operatorname {PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{9} b n x^3 \operatorname {PolyLog}(3,e x)\)

\(\Big \downarrow \) 7145

\(\displaystyle \frac {1}{9} b n \left (\frac {1}{3} x^3 \operatorname {PolyLog}(2,e x)-\frac {1}{3} \int -x^2 \log (1-e x)dx\right )+\frac {1}{3} \left (\frac {1}{3} \left (\frac {\log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{3 e^2}-\frac {1}{3} x^3 \log (1-e x) \left (a+b \log \left (c x^n\right )\right )+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{6 e}+\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )+b n \left (\frac {\operatorname {PolyLog}(2,e x)}{3 e^3}-\frac {\log (1-e x)}{9 e^3}-\frac {4 x}{9 e^2}+\frac {1}{9} x^3 \log (1-e x)-\frac {5 x^2}{36 e}-\frac {2 x^3}{27}\right )\right )-\frac {1}{3} x^3 \operatorname {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )+\frac {1}{9} b n \left (\frac {1}{3} e \left (-\frac {\log (1-e x)}{e^4}-\frac {x}{e^3}-\frac {x^2}{2 e^2}-\frac {x^3}{3 e}\right )+\frac {1}{3} x^3 \log (1-e x)\right )+\frac {1}{9} b n x^3 \operatorname {PolyLog}(2,e x)\right )+\frac {1}{3} x^3 \operatorname {PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{9} b n x^3 \operatorname {PolyLog}(3,e x)\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{9} b n \left (\frac {1}{3} \int x^2 \log (1-e x)dx+\frac {1}{3} x^3 \operatorname {PolyLog}(2,e x)\right )+\frac {1}{3} \left (\frac {1}{3} \left (\frac {\log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{3 e^2}-\frac {1}{3} x^3 \log (1-e x) \left (a+b \log \left (c x^n\right )\right )+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{6 e}+\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )+b n \left (\frac {\operatorname {PolyLog}(2,e x)}{3 e^3}-\frac {\log (1-e x)}{9 e^3}-\frac {4 x}{9 e^2}+\frac {1}{9} x^3 \log (1-e x)-\frac {5 x^2}{36 e}-\frac {2 x^3}{27}\right )\right )-\frac {1}{3} x^3 \operatorname {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )+\frac {1}{9} b n \left (\frac {1}{3} e \left (-\frac {\log (1-e x)}{e^4}-\frac {x}{e^3}-\frac {x^2}{2 e^2}-\frac {x^3}{3 e}\right )+\frac {1}{3} x^3 \log (1-e x)\right )+\frac {1}{9} b n x^3 \operatorname {PolyLog}(2,e x)\right )+\frac {1}{3} x^3 \operatorname {PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{9} b n x^3 \operatorname {PolyLog}(3,e x)\)

\(\Big \downarrow \) 2842

\(\displaystyle \frac {1}{9} b n \left (\frac {1}{3} \left (\frac {1}{3} e \int \frac {x^3}{1-e x}dx+\frac {1}{3} x^3 \log (1-e x)\right )+\frac {1}{3} x^3 \operatorname {PolyLog}(2,e x)\right )+\frac {1}{3} \left (\frac {1}{3} \left (\frac {\log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{3 e^2}-\frac {1}{3} x^3 \log (1-e x) \left (a+b \log \left (c x^n\right )\right )+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{6 e}+\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )+b n \left (\frac {\operatorname {PolyLog}(2,e x)}{3 e^3}-\frac {\log (1-e x)}{9 e^3}-\frac {4 x}{9 e^2}+\frac {1}{9} x^3 \log (1-e x)-\frac {5 x^2}{36 e}-\frac {2 x^3}{27}\right )\right )-\frac {1}{3} x^3 \operatorname {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )+\frac {1}{9} b n \left (\frac {1}{3} e \left (-\frac {\log (1-e x)}{e^4}-\frac {x}{e^3}-\frac {x^2}{2 e^2}-\frac {x^3}{3 e}\right )+\frac {1}{3} x^3 \log (1-e x)\right )+\frac {1}{9} b n x^3 \operatorname {PolyLog}(2,e x)\right )+\frac {1}{3} x^3 \operatorname {PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{9} b n x^3 \operatorname {PolyLog}(3,e x)\)

\(\Big \downarrow \) 49

\(\displaystyle \frac {1}{9} b n \left (\frac {1}{3} \left (\frac {1}{3} e \int \left (-\frac {x^2}{e}-\frac {x}{e^2}-\frac {1}{e^3 (e x-1)}-\frac {1}{e^3}\right )dx+\frac {1}{3} x^3 \log (1-e x)\right )+\frac {1}{3} x^3 \operatorname {PolyLog}(2,e x)\right )+\frac {1}{3} \left (\frac {1}{3} \left (\frac {\log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{3 e^2}-\frac {1}{3} x^3 \log (1-e x) \left (a+b \log \left (c x^n\right )\right )+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{6 e}+\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )+b n \left (\frac {\operatorname {PolyLog}(2,e x)}{3 e^3}-\frac {\log (1-e x)}{9 e^3}-\frac {4 x}{9 e^2}+\frac {1}{9} x^3 \log (1-e x)-\frac {5 x^2}{36 e}-\frac {2 x^3}{27}\right )\right )-\frac {1}{3} x^3 \operatorname {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )+\frac {1}{9} b n \left (\frac {1}{3} e \left (-\frac {\log (1-e x)}{e^4}-\frac {x}{e^3}-\frac {x^2}{2 e^2}-\frac {x^3}{3 e}\right )+\frac {1}{3} x^3 \log (1-e x)\right )+\frac {1}{9} b n x^3 \operatorname {PolyLog}(2,e x)\right )+\frac {1}{3} x^3 \operatorname {PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{9} b n x^3 \operatorname {PolyLog}(3,e x)\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{3} \left (\frac {1}{3} \left (\frac {\log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{3 e^2}-\frac {1}{3} x^3 \log (1-e x) \left (a+b \log \left (c x^n\right )\right )+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{6 e}+\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )+b n \left (\frac {\operatorname {PolyLog}(2,e x)}{3 e^3}-\frac {\log (1-e x)}{9 e^3}-\frac {4 x}{9 e^2}+\frac {1}{9} x^3 \log (1-e x)-\frac {5 x^2}{36 e}-\frac {2 x^3}{27}\right )\right )-\frac {1}{3} x^3 \operatorname {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )+\frac {1}{9} b n \left (\frac {1}{3} e \left (-\frac {\log (1-e x)}{e^4}-\frac {x}{e^3}-\frac {x^2}{2 e^2}-\frac {x^3}{3 e}\right )+\frac {1}{3} x^3 \log (1-e x)\right )+\frac {1}{9} b n x^3 \operatorname {PolyLog}(2,e x)\right )+\frac {1}{3} x^3 \operatorname {PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )+\frac {1}{9} b n \left (\frac {1}{3} \left (\frac {1}{3} e \left (-\frac {\log (1-e x)}{e^4}-\frac {x}{e^3}-\frac {x^2}{2 e^2}-\frac {x^3}{3 e}\right )+\frac {1}{3} x^3 \log (1-e x)\right )+\frac {1}{3} x^3 \operatorname {PolyLog}(2,e x)\right )-\frac {1}{9} b n x^3 \operatorname {PolyLog}(3,e x)\)

Input:

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

Output:

(b*n*(((x^3*Log[1 - e*x])/3 + (e*(-(x/e^3) - x^2/(2*e^2) - x^3/(3*e) - Log 
[1 - e*x]/e^4))/3)/3 + (x^3*PolyLog[2, e*x])/3))/9 + ((b*n*((x^3*Log[1 - e 
*x])/3 + (e*(-(x/e^3) - x^2/(2*e^2) - x^3/(3*e) - Log[1 - e*x]/e^4))/3))/9 
 + (b*n*x^3*PolyLog[2, e*x])/9 - (x^3*(a + b*Log[c*x^n])*PolyLog[2, e*x])/ 
3 + ((x*(a + b*Log[c*x^n]))/(3*e^2) + (x^2*(a + b*Log[c*x^n]))/(6*e) + (x^ 
3*(a + b*Log[c*x^n]))/9 + ((a + b*Log[c*x^n])*Log[1 - e*x])/(3*e^3) - (x^3 
*(a + b*Log[c*x^n])*Log[1 - e*x])/3 + b*n*((-4*x)/(9*e^2) - (5*x^2)/(36*e) 
 - (2*x^3)/27 - Log[1 - e*x]/(9*e^3) + (x^3*Log[1 - e*x])/9 + PolyLog[2, e 
*x]/(3*e^3)))/3)/3 - (b*n*x^3*PolyLog[3, e*x])/9 + (x^3*(a + b*Log[c*x^n]) 
*PolyLog[3, e*x])/3
 

Defintions of rubi rules used

rule 25

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

rule 49

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] 
&& IGtQ[m, 0] && IGtQ[m + n + 2, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2823

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_. 
)]*(b_.))*((g_.)*(x_))^(q_.), x_Symbol] :> With[{u = IntHide[(g*x)^q*Log[d* 
(e + f*x^m)^r], x]}, Simp[(a + b*Log[c*x^n])   u, x] - Simp[b*n   Int[1/x 
 u, x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, q}, x] && (IntegerQ[(q 
+ 1)/m] || (RationalQ[m] && RationalQ[q])) && NeQ[q, -1]
 

rule 2832

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.)*PolyLog[k_, (e 
_.)*(x_)^(q_.)], x_Symbol] :> Simp[(-b)*n*(d*x)^(m + 1)*(PolyLog[k, e*x^q]/ 
(d*(m + 1)^2)), x] + (Simp[(d*x)^(m + 1)*PolyLog[k, e*x^q]*((a + b*Log[c*x^ 
n])/(d*(m + 1))), x] - Simp[q/(m + 1)   Int[(d*x)^m*PolyLog[k - 1, e*x^q]*( 
a + b*Log[c*x^n]), x], x] + Simp[b*n*(q/(m + 1)^2)   Int[(d*x)^m*PolyLog[k 
- 1, e*x^q], x], x]) /; FreeQ[{a, b, c, d, e, m, n, q}, x] && IGtQ[k, 0]
 

rule 2842

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

rule 7145

Int[((d_.)*(x_))^(m_.)*PolyLog[n_, (a_.)*((b_.)*(x_)^(p_.))^(q_.)], x_Symbo 
l] :> Simp[(d*x)^(m + 1)*(PolyLog[n, a*(b*x^p)^q]/(d*(m + 1))), x] - Simp[p 
*(q/(m + 1))   Int[(d*x)^m*PolyLog[n - 1, a*(b*x^p)^q], x], x] /; FreeQ[{a, 
 b, d, m, p, q}, x] && NeQ[m, -1] && GtQ[n, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.17 \[ \int x^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(3,e x) \, dx=-\frac {4 \, {\left (4 \, b e^{3} n - 3 \, a e^{3}\right )} x^{3} + 9 \, {\left (3 \, b e^{2} n - 2 \, a e^{2}\right )} x^{2} + 36 \, {\left (2 \, b e n - a e\right )} x - 36 \, {\left ({\left (2 \, b e^{3} n - 3 \, a e^{3}\right )} x^{3} + b n\right )} {\rm Li}_2\left (e x\right ) - 36 \, {\left ({\left (b e^{3} n - a e^{3}\right )} x^{3} - b n + a\right )} \log \left (-e x + 1\right ) + 6 \, {\left (18 \, b e^{3} x^{3} {\rm Li}_2\left (e x\right ) - 2 \, b e^{3} x^{3} - 3 \, b e^{2} x^{2} - 6 \, b e x + 6 \, {\left (b e^{3} x^{3} - b\right )} \log \left (-e x + 1\right )\right )} \log \left (c\right ) + 6 \, {\left (18 \, b e^{3} n x^{3} {\rm Li}_2\left (e x\right ) - 2 \, b e^{3} n x^{3} - 3 \, b e^{2} n x^{2} - 6 \, b e n x + 6 \, {\left (b e^{3} n x^{3} - b n\right )} \log \left (-e x + 1\right )\right )} \log \left (x\right ) - 108 \, {\left (3 \, b e^{3} n x^{3} \log \left (x\right ) + 3 \, b e^{3} x^{3} \log \left (c\right ) - {\left (b e^{3} n - 3 \, a e^{3}\right )} x^{3}\right )} {\rm polylog}\left (3, e x\right )}{972 \, e^{3}} \] Input:

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

Output:

-1/972*(4*(4*b*e^3*n - 3*a*e^3)*x^3 + 9*(3*b*e^2*n - 2*a*e^2)*x^2 + 36*(2* 
b*e*n - a*e)*x - 36*((2*b*e^3*n - 3*a*e^3)*x^3 + b*n)*dilog(e*x) - 36*((b* 
e^3*n - a*e^3)*x^3 - b*n + a)*log(-e*x + 1) + 6*(18*b*e^3*x^3*dilog(e*x) - 
 2*b*e^3*x^3 - 3*b*e^2*x^2 - 6*b*e*x + 6*(b*e^3*x^3 - b)*log(-e*x + 1))*lo 
g(c) + 6*(18*b*e^3*n*x^3*dilog(e*x) - 2*b*e^3*n*x^3 - 3*b*e^2*n*x^2 - 6*b* 
e*n*x + 6*(b*e^3*n*x^3 - b*n)*log(-e*x + 1))*log(x) - 108*(3*b*e^3*n*x^3*l 
og(x) + 3*b*e^3*x^3*log(c) - (b*e^3*n - 3*a*e^3)*x^3)*polylog(3, e*x))/e^3
 

Sympy [F]

\[ \int x^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(3,e x) \, dx=\int x^{2} \left (a + b \log {\left (c x^{n} \right )}\right ) \operatorname {Li}_{3}\left (e x\right )\, dx \] Input:

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

Output:

Integral(x**2*(a + b*log(c*x**n))*polylog(3, e*x), x)
 

Maxima [F]

\[ \int x^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(3,e x) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} x^{2} {\rm Li}_{3}(e x) \,d x } \] Input:

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

Output:

-1/162*b*((6*(3*e^3*x^3*log(x^n) - (2*e^3*n - 3*e^3*log(c))*x^3)*dilog(e*x 
) - 6*((e^3*n - e^3*log(c))*x^3 - n*log(x))*log(-e*x + 1) - (2*e^3*x^3 + 3 
*e^2*x^2 + 6*e*x - 6*(e^3*x^3 - 1)*log(-e*x + 1))*log(x^n) - 18*(3*e^3*x^3 
*log(x^n) - (e^3*n - 3*e^3*log(c))*x^3)*polylog(3, e*x))/e^3 - 162*integra 
te(-1/162*(e^2*n*x^2 + 2*(4*e^3*n - 3*e^3*log(c))*x^3 + 3*e*n*x - 6*n*log( 
x) - 6*n)/(e^3*x - e^2), x)) - 1/162*(18*e^3*x^3*dilog(e*x) - 54*e^3*x^3*p 
olylog(3, e*x) - 2*e^3*x^3 - 3*e^2*x^2 - 6*e*x + 6*(e^3*x^3 - 1)*log(-e*x 
+ 1))*a/e^3
 

Giac [F]

\[ \int x^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(3,e x) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} x^{2} {\rm Li}_{3}(e x) \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

\text{Hanged}
 

Reduce [F]

\[ \int x^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(3,e x) \, dx=\left (\int \mathrm {log}\left (x^{n} c \right ) \mathit {polylog}\left (3, e x \right ) x^{2}d x \right ) b +\left (\int \mathit {polylog}\left (3, e x \right ) x^{2}d x \right ) a \] Input:

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

Output:

int(log(x**n*c)*polylog(3,e*x)*x**2,x)*b + int(polylog(3,e*x)*x**2,x)*a