\(\int x (a+b \log (c x^n)) \operatorname {PolyLog}(3,e x) \, dx\) [221]

Optimal result

Integrand size = 17, antiderivative size = 221 \[ \int x \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(3,e x) \, dx=-\frac {5 b n x}{16 e}-\frac {1}{8} b n x^2+\frac {x \left (a+b \log \left (c x^n\right )\right )}{8 e}+\frac {1}{16} x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {3 b n \log (1-e x)}{16 e^2}+\frac {3}{16} b n x^2 \log (1-e x)+\frac {\left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{8 e^2}-\frac {1}{8} x^2 \left (a+b \log \left (c x^n\right )\right ) \log (1-e x)+\frac {b n \operatorname {PolyLog}(2,e x)}{8 e^2}+\frac {1}{4} b n x^2 \operatorname {PolyLog}(2,e x)-\frac {1}{4} x^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(2,e x)-\frac {1}{4} b n x^2 \operatorname {PolyLog}(3,e x)+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(3,e x) \] Output:

-5/16*b*n*x/e-1/8*b*n*x^2+1/8*x*(a+b*ln(c*x^n))/e+1/16*x^2*(a+b*ln(c*x^n)) 
-3/16*b*n*ln(-e*x+1)/e^2+3/16*b*n*x^2*ln(-e*x+1)+1/8*(a+b*ln(c*x^n))*ln(-e 
*x+1)/e^2-1/8*x^2*(a+b*ln(c*x^n))*ln(-e*x+1)+1/8*b*n*polylog(2,e*x)/e^2+1/ 
4*b*n*x^2*polylog(2,e*x)-1/4*x^2*(a+b*ln(c*x^n))*polylog(2,e*x)-1/4*b*n*x^ 
2*polylog(3,e*x)+1/2*x^2*(a+b*ln(c*x^n))*polylog(3,e*x)
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 1.07 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.59, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.765, 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.

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

\(\Big \downarrow \) 2832

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

\(\Big \downarrow \) 2832

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2823

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 2842

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

\(\Big \downarrow \) 49

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 7145

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2842

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

\(\Big \downarrow \) 49

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.16 \[ \int x \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}(3,e x) \, dx=-\frac {{\left (2 \, b e^{2} n - a e^{2}\right )} x^{2} + {\left (5 \, b e n - 2 \, a e\right )} x - 2 \, {\left (2 \, {\left (b e^{2} n - a e^{2}\right )} x^{2} + b n\right )} {\rm Li}_2\left (e x\right ) - {\left ({\left (3 \, b e^{2} n - 2 \, a e^{2}\right )} x^{2} - 3 \, b n + 2 \, a\right )} \log \left (-e x + 1\right ) + {\left (4 \, b e^{2} x^{2} {\rm Li}_2\left (e x\right ) - b e^{2} x^{2} - 2 \, b e x + 2 \, {\left (b e^{2} x^{2} - b\right )} \log \left (-e x + 1\right )\right )} \log \left (c\right ) + {\left (4 \, b e^{2} n x^{2} {\rm Li}_2\left (e x\right ) - b e^{2} n x^{2} - 2 \, b e n x + 2 \, {\left (b e^{2} n x^{2} - b n\right )} \log \left (-e x + 1\right )\right )} \log \left (x\right ) - 4 \, {\left (2 \, b e^{2} n x^{2} \log \left (x\right ) + 2 \, b e^{2} x^{2} \log \left (c\right ) - {\left (b e^{2} n - 2 \, a e^{2}\right )} x^{2}\right )} {\rm polylog}\left (3, e x\right )}{16 \, e^{2}} \] Input:

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

Output:

-1/16*((2*b*e^2*n - a*e^2)*x^2 + (5*b*e*n - 2*a*e)*x - 2*(2*(b*e^2*n - a*e 
^2)*x^2 + b*n)*dilog(e*x) - ((3*b*e^2*n - 2*a*e^2)*x^2 - 3*b*n + 2*a)*log( 
-e*x + 1) + (4*b*e^2*x^2*dilog(e*x) - b*e^2*x^2 - 2*b*e*x + 2*(b*e^2*x^2 - 
 b)*log(-e*x + 1))*log(c) + (4*b*e^2*n*x^2*dilog(e*x) - b*e^2*n*x^2 - 2*b* 
e*n*x + 2*(b*e^2*n*x^2 - b*n)*log(-e*x + 1))*log(x) - 4*(2*b*e^2*n*x^2*log 
(x) + 2*b*e^2*x^2*log(c) - (b*e^2*n - 2*a*e^2)*x^2)*polylog(3, e*x))/e^2
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-1/16*b*((4*(e^2*x^2*log(x^n) - (e^2*n - e^2*log(c))*x^2)*dilog(e*x) - ((3 
*e^2*n - 2*e^2*log(c))*x^2 - 2*n*log(x))*log(-e*x + 1) - (e^2*x^2 + 2*e*x 
- 2*(e^2*x^2 - 1)*log(-e*x + 1))*log(x^n) - 4*(2*e^2*x^2*log(x^n) - (e^2*n 
 - 2*e^2*log(c))*x^2)*polylog(3, e*x))/e^2 - 16*integrate(-1/16*(e*n*x + 2 
*(2*e^2*n - e^2*log(c))*x^2 - 2*n*log(x) - 2*n)/(e^2*x - e), x)) - 1/16*(4 
*e^2*x^2*dilog(e*x) - 8*e^2*x^2*polylog(3, e*x) - e^2*x^2 - 2*e*x + 2*(e^2 
*x^2 - 1)*log(-e*x + 1))*a/e^2
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

\text{Hanged}
 

Reduce [F]

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

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

Output:

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