3.2.82 \(\int \frac {(g+h \log (f (d+e x)^n)) \operatorname {PolyLog}(2,c (a+b x))}{x^3} \, dx\) [182]

3.2.82.1 Optimal result

Integrand size = 27, antiderivative size = 3119 \[ \int \frac {\left (g+h \log \left (f (d+e x)^n\right )\right ) \operatorname {PolyLog}(2,c (a+b x))}{x^3} \, dx =\text {Too large to display} \]

output

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

3.2.82.2 Mathematica [A] (verified)

Time = 13.02 (sec) , antiderivative size = 2700, normalized size of antiderivative = 0.87 \[ \int \frac {\left (g+h \log \left (f (d+e x)^n\right )\right ) \operatorname {PolyLog}(2,c (a+b x))}{x^3} \, dx=\text {Result too large to show} \]

input

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

output

((g - h*n*Log[d + e*x] + h*Log[f*(d + e*x)^n])*(-((-1 + a*c)*(a^2 - b^2*x^ 
2)*PolyLog[2, c*(a + b*x)]) + b*x*(-(a*b*c*x*Log[x]) + (a*(-1 + a*c + b*c* 
x) + b*(-1 + a*c)*x*Log[(b*c*x)/(1 - a*c)])*Log[1 - a*c - b*c*x] + b*(-1 + 
 a*c)*x*PolyLog[2, (-1 + a*c + b*c*x)/(-1 + a*c)])))/(2*a^2*(-1 + a*c)*x^2 
) - (h*n*(((d*e*x + e^2*x^2*Log[x] + (d^2 - e^2*x^2)*Log[d + e*x])*PolyLog 
[2, c*(a + b*x)])/x^2 + (b*d*e*(Log[x]*Log[1 - a*c - b*c*x] - Log[c*(a + b 
*x)]*Log[1 - a*c - b*c*x] - Log[x]*Log[1 + (b*c*x)/(-1 + a*c)] - PolyLog[2 
, (b*c*x)/(1 - a*c)] - PolyLog[2, 1 - a*c - b*c*x]))/a + e^2*(Log[x]*Log[1 
 + (b*x)/a]*Log[1 - a*c - b*c*x] + ((-Log[c*(a + b*x)] + Log[1 + (b*x)/a]) 
*Log[1 - a*c - b*c*x]*(-2*Log[x] + Log[1 - a*c - b*c*x]))/2 + (Log[c*(a + 
b*x)] - Log[1 + (b*x)/a])*Log[1 - a*c - b*c*x]*Log[(a*(-1 + a*c + b*c*x))/ 
(b*x)] + ((Log[(1 - a*c)/(b*c*x)] - Log[((1 - a*c)*(a + b*x))/(b*x)] + Log 
[1 + (b*x)/a])*Log[(a*(-1 + a*c + b*c*x))/(b*x)]^2)/2 + (Log[1 - a*c - b*c 
*x] - Log[(a*(-1 + a*c + b*c*x))/(b*x)])*PolyLog[2, -((b*x)/a)] + (Log[x] 
+ Log[(a*(-1 + a*c + b*c*x))/(b*x)])*PolyLog[2, 1 - a*c - b*c*x] + Log[(a* 
(-1 + a*c + b*c*x))/(b*x)]*(-PolyLog[2, (a*(-1 + a*c + b*c*x))/(b*x)] + Po 
lyLog[2, (-1 + a*c + b*c*x)/(b*c*x)]) - PolyLog[3, -((b*x)/a)] - PolyLog[3 
, 1 - a*c - b*c*x] + PolyLog[3, (a*(-1 + a*c + b*c*x))/(b*x)] - PolyLog[3, 
 (-1 + a*c + b*c*x)/(b*c*x)]) - e^2*(Log[c*(a + b*x)]*Log[1 - a*c - b*c*x] 
*Log[d + e*x] + ((Log[c*(a + b*x)] - Log[(e*(a + b*x))/(-(b*d) + a*e)])...
 

3.2.82.3 Rubi [A] (verified)

Time = 4.49 (sec) , antiderivative size = 2903, normalized size of antiderivative = 0.93, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {7157, 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.

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

\(\Big \downarrow \) 7157

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

\(\Big \downarrow \) 2009

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

input

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

output

-1/2*((g + h*Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)])/x^2 + (e*h*n*(-( 
(b*Log[(b*c*x)/(1 - a*c)]*Log[1 - a*c - b*c*x])/(a*d)) + (e*(Log[c*(a + b* 
x)] + Log[(b*c*d + e - a*c*e)/(b*c*(d + e*x))] - Log[((b*c*d + e - a*c*e)* 
(a + b*x))/(b*(d + e*x))])*Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x) 
))]^2)/(2*d^2) - (e*Log[x]*Log[1 + (b*x)/a]*Log[1 - c*(a + b*x)])/d^2 + (e 
*Log[c*(a + b*x)]*Log[d + e*x]*Log[1 - c*(a + b*x)])/d^2 - (e*(Log[c*(a + 
b*x)] - Log[-((e*(a + b*x))/(b*d - a*e))])*(Log[(b*(d + e*x))/((b*d - a*e) 
*(1 - c*(a + b*x)))] + Log[1 - c*(a + b*x)])^2)/(2*d^2) - (e*(Log[1 + (b*x 
)/a] + Log[(1 - a*c)/(1 - c*(a + b*x))] - Log[((1 - a*c)*(a + b*x))/(a*(1 
- c*(a + b*x)))])*Log[-((a*(1 - c*(a + b*x)))/(b*x))]^2)/(2*d^2) - (e*(Log 
[c*(a + b*x)] - Log[1 + (b*x)/a])*(Log[x] + Log[-((a*(1 - c*(a + b*x)))/(b 
*x))])^2)/(2*d^2) - (e*(Log[1 - c*(a + b*x)] - Log[-((a*(1 - c*(a + b*x))) 
/(b*x))])*PolyLog[2, -((b*x)/a)])/d^2 - (b*PolyLog[2, c*(a + b*x)])/(a*d) 
- PolyLog[2, c*(a + b*x)]/(d*x) - (e*Log[x]*PolyLog[2, c*(a + b*x)])/d^2 + 
 (e*Log[d + e*x]*PolyLog[2, c*(a + b*x)])/d^2 - (b*PolyLog[2, 1 - (b*c*x)/ 
(1 - a*c)])/(a*d) + (e*(Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + b*x)))] 
 + Log[1 - c*(a + b*x)])*PolyLog[2, (b*(d + e*x))/(b*d - a*e)])/d^2 - (e*L 
og[-((a*(1 - c*(a + b*x)))/(b*x))]*PolyLog[2, -((b*x)/(a*(1 - c*(a + b*x)) 
))])/d^2 + (e*Log[-((a*(1 - c*(a + b*x)))/(b*x))]*PolyLog[2, -((b*c*x)/(1 
- c*(a + b*x)))])/d^2 + (e*(Log[d + e*x] - Log[(b*(d + e*x))/((b*d - a*...
 

3.2.82.3.1 Defintions of rubi rules used

rule 2009

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

rule 7157

Int[((g_.) + Log[(f_.)*((d_.) + (e_.)*(x_))^(n_.)]*(h_.))*(x_)^(m_.)*PolyLo 
g[2, (c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> Simp[x^(m + 1)*(g + h*Log[f* 
(d + e*x)^n])*(PolyLog[2, c*(a + b*x)]/(m + 1)), x] + (Simp[b/(m + 1)   Int 
[ExpandIntegrand[(g + h*Log[f*(d + e*x)^n])*Log[1 - a*c - b*c*x], x^(m + 1) 
/(a + b*x), x], x], x] - Simp[e*h*(n/(m + 1))   Int[ExpandIntegrand[PolyLog 
[2, c*(a + b*x)], x^(m + 1)/(d + e*x), x], x], x]) /; FreeQ[{a, b, c, d, e, 
 f, g, h, n}, x] && IntegerQ[m] && NeQ[m, -1]
 

3.2.82.4 Maple [F]

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

input

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

output

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

3.2.82.5 Fricas [F]

\[ \int \frac {\left (g+h \log \left (f (d+e x)^n\right )\right ) \operatorname {PolyLog}(2,c (a+b x))}{x^3} \, dx=\int { \frac {{\left (h \log \left ({\left (e x + d\right )}^{n} f\right ) + g\right )} {\rm Li}_2\left ({\left (b x + a\right )} c\right )}{x^{3}} \,d x } \]

input

integrate((g+h*log(f*(e*x+d)^n))*polylog(2,c*(b*x+a))/x^3,x, algorithm="fr 
icas")
 

output

integral((h*dilog(b*c*x + a*c)*log((e*x + d)^n*f) + g*dilog(b*c*x + a*c))/ 
x^3, x)
 

3.2.82.6 Sympy [F(-1)]

Timed out. \[ \int \frac {\left (g+h \log \left (f (d+e x)^n\right )\right ) \operatorname {PolyLog}(2,c (a+b x))}{x^3} \, dx=\text {Timed out} \]

input

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

output

Timed out
 

3.2.82.7 Maxima [F]

\[ \int \frac {\left (g+h \log \left (f (d+e x)^n\right )\right ) \operatorname {PolyLog}(2,c (a+b x))}{x^3} \, dx=\int { \frac {{\left (h \log \left ({\left (e x + d\right )}^{n} f\right ) + g\right )} {\rm Li}_2\left ({\left (b x + a\right )} c\right )}{x^{3}} \,d x } \]

input

integrate((g+h*log(f*(e*x+d)^n))*polylog(2,c*(b*x+a))/x^3,x, algorithm="ma 
xima")
 

output

integrate((h*log((e*x + d)^n*f) + g)*dilog((b*x + a)*c)/x^3, x)
 

3.2.82.8 Giac [F]

\[ \int \frac {\left (g+h \log \left (f (d+e x)^n\right )\right ) \operatorname {PolyLog}(2,c (a+b x))}{x^3} \, dx=\int { \frac {{\left (h \log \left ({\left (e x + d\right )}^{n} f\right ) + g\right )} {\rm Li}_2\left ({\left (b x + a\right )} c\right )}{x^{3}} \,d x } \]

input

integrate((g+h*log(f*(e*x+d)^n))*polylog(2,c*(b*x+a))/x^3,x, algorithm="gi 
ac")
 

output

integrate((h*log((e*x + d)^n*f) + g)*dilog((b*x + a)*c)/x^3, x)
 

3.2.82.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (g+h \log \left (f (d+e x)^n\right )\right ) \operatorname {PolyLog}(2,c (a+b x))}{x^3} \, dx=\int \frac {\mathrm {polylog}\left (2,c\,\left (a+b\,x\right )\right )\,\left (g+h\,\ln \left (f\,{\left (d+e\,x\right )}^n\right )\right )}{x^3} \,d x \]

input

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

output

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