3.2.78 \(\int x (g+h \log (f (d+e x)^n)) \operatorname {PolyLog}(2,c (a+b x)) \, dx\) [178]

3.2.78.1 Optimal result

Integrand size = 25, antiderivative size = 2252 \[ \int x \left (g+h \log \left (f (d+e x)^n\right )\right ) \operatorname {PolyLog}(2,c (a+b x)) \, dx =\text {Too large to display} \]

output

-1/4*h*n*x^2*ln(-b*c*x-a*c+1)+1/2*a*g*x/b-1/2*a^2*g*polylog(2,c*(b*x+a))/b 
^2-1/4*h*n*x^2*polylog(2,c*(b*x+a))-1/4*d^2*h*n*polylog(2,e*(-b*c*x-a*c+1) 
/(-a*c*e+b*c*d+e))/e^2-1/2*a^2*h*n*polylog(3,b*(e*x+d)/(-a*e+b*d))/b^2+1/2 
*d^2*h*n*polylog(3,b*(e*x+d)/(-a*e+b*d))/e^2-1/2*a^2*h*n*polylog(3,1-c*(b* 
x+a))/b^2+1/2*d^2*h*n*polylog(3,1-c*(b*x+a))/e^2-1/2*a^2*h*n*polylog(3,-e* 
(1-c*(b*x+a))/b/c/(e*x+d))/b^2+1/2*d^2*h*n*polylog(3,-e*(1-c*(b*x+a))/b/c/ 
(e*x+d))/e^2+1/2*a^2*h*n*polylog(3,(-a*e+b*d)*(1-c*(b*x+a))/b/(e*x+d))/b^2 
-1/2*d^2*h*n*polylog(3,(-a*e+b*d)*(1-c*(b*x+a))/b/(e*x+d))/e^2-1/8*x^2*(g+ 
h*ln(f*(e*x+d)^n))-7/8*d*h*n*x/e+1/4*a^2*h*n*polylog(2,c*(b*x+a))/b^2+1/2* 
x^2*(g+h*ln(f*(e*x+d)^n))*polylog(2,c*(b*x+a))+3/16*h*n*x^2+1/4*x^2*ln(-b* 
c*x-a*c+1)*(g+h*ln(f*(e*x+d)^n))+1/8*d^2*h*n*ln(e*x+d)/e^2-3/4*a*h*n*(-b*c 
*x-a*c+1)*ln(-b*c*x-a*c+1)/b^2/c-1/4*(-a*c+1)*h*(e*x+d)*ln(f*(e*x+d)^n)/b/ 
c/e+1/2*a^2*h*n*ln(c*(b*x+a))*ln(e*x+d)*ln(1-c*(b*x+a))/b^2-1/2*d^2*h*n*ln 
(c*(b*x+a))*ln(e*x+d)*ln(1-c*(b*x+a))/e^2+1/2*a*d*h*n*polylog(2,c*(b*x+a)) 
/b/e-1/2*a*d*h*n*polylog(2,e*(-b*c*x-a*c+1)/(-a*c*e+b*c*d+e))/b/e+1/2*a*(- 
a*c+1)*h*n*polylog(2,b*c*(e*x+d)/(-a*c*e+b*c*d+e))/b^2/c+1/4*d^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/e^2-1/2*d^2*h*n*ln(e*x+d)*polylog(2,c*(b*x+a))/e^2+1 
/2*a^2*h*n*(ln(b*(e*x+d)/(-a*e+b*d)/(1-c*(b*x+a)))+ln(1-c*(b*x+a)))*polylo 
g(2,b*(e*x+d)/(-a*e+b*d))/b^2-1/4*(-a*c+1)^2*h*n*polylog(2,b*c*(e*x+d)/...
 

3.2.78.2 Mathematica [A] (verified)

Time = 5.25 (sec) , antiderivative size = 1996, normalized size of antiderivative = 0.89 \[ \int x \left (g+h \log \left (f (d+e x)^n\right )\right ) \operatorname {PolyLog}(2,c (a+b x)) \, dx =\text {Too large to display} \]

input

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

output

((g - h*n*Log[d + e*x] + h*Log[f*(d + e*x)^n])*(-(b*c*x*(2 - 6*a*c + b*c*x 
)) + (-2 - 6*a^2*c^2 + 2*b^2*c^2*x^2 - 4*a*c*(-2 + b*c*x))*Log[1 - a*c - b 
*c*x] - 4*c^2*(a^2 - b^2*x^2)*PolyLog[2, c*(a + b*x)]))/(8*b^2*c^2) + (h*n 
*(4*b^2*c^2*(e*x*(2*d - e*x) - 2*(d^2 - e^2*x^2)*Log[d + e*x])*PolyLog[2, 
c*(a + b*x)] + 8*b*c*d*e*(1 - a*c - b*c*x + (-1 + a*c + b*c*x - a*c*Log[c* 
(a + b*x)])*Log[1 - a*c - b*c*x] - a*c*PolyLog[2, 1 - a*c - b*c*x]) + e^2* 
(c*(-4*a^2*c + a*(4 - 6*b*c*x) + b*x*(2 + b*c*x)) + (2 + 6*a^2*c^2 - 2*b^2 
*c^2*x^2 + 4*a*c*(-2 + b*c*x) - 4*a^2*c^2*Log[c*(a + b*x)])*Log[1 - a*c - 
b*c*x] - 4*a^2*c^2*PolyLog[2, 1 - a*c - b*c*x]) - 8*b^2*c^2*d^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)])*Log[(b*(d + e*x))/(b*d - a*e)]*(-2*Log[1 - a*c - 
 b*c*x] + Log[(b*(d + e*x))/(b*d - a*e)]))/2 + (-Log[c*(a + b*x)] + Log[(e 
*(a + b*x))/(-(b*d) + a*e)])*Log[(b*(d + e*x))/(b*d - a*e)]*Log[-((b*(d + 
e*x))/((b*d - a*e)*(-1 + a*c + b*c*x)))] + (Log[-((b*(d + e*x))/((b*d - a* 
e)*(-1 + a*c + b*c*x)))]^2*(Log[c*(a + b*x)] - Log[((b*c*d + e - a*c*e)*(a 
 + b*x))/((b*d - a*e)*(-1 + a*c + b*c*x))] + Log[(b*c*d + e - a*c*e)/(e - 
a*c*e - b*c*e*x)]))/2 + (Log[d + e*x] - Log[-((b*(d + e*x))/((b*d - a*e)*( 
-1 + a*c + b*c*x)))])*PolyLog[2, 1 - a*c - b*c*x] + (Log[1 - a*c - b*c*x] 
+ Log[-((b*(d + e*x))/((b*d - a*e)*(-1 + a*c + b*c*x)))])*PolyLog[2, (b*(d 
 + e*x))/(b*d - a*e)] + Log[-((b*(d + e*x))/((b*d - a*e)*(-1 + a*c + b*...
 

3.2.78.3 Rubi [A] (verified)

Time = 3.77 (sec) , antiderivative size = 2358, normalized size of antiderivative = 1.05, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, 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.

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

\(\Big \downarrow \) 7157

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (g+h \log \left (f (d+e x)^n\right )\right ) \operatorname {PolyLog}(2,c (a+b x)) x^2-\frac {1}{2} e h n \left (-\frac {\operatorname {PolyLog}(2,c (a+b x)) a^2}{2 b^2 e}+\frac {x a}{2 b e}+\frac {(-a c-b x c+1) \log (-a c-b x c+1) a}{2 b^2 c e}-\frac {d \operatorname {PolyLog}(2,c (a+b x)) a}{b e^2}-\frac {x^2}{8 e}+\frac {d^2 \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 e^3}-\frac {d^2 \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 e^3}-\frac {(1-a c) x}{4 b c e}+\frac {d x}{e^2}+\frac {x^2 \log (-a c-b x c+1)}{4 e}+\frac {d (-a c-b x c+1) \log (-a c-b x c+1)}{b c e^2}-\frac {(1-a c)^2 \log (-a c-b x c+1)}{4 b^2 c^2 e}+\frac {d^2 \log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{e^3}+\frac {x^2 \operatorname {PolyLog}(2,c (a+b x))}{2 e}-\frac {d x \operatorname {PolyLog}(2,c (a+b x))}{e^2}+\frac {d^2 \log (d+e x) \operatorname {PolyLog}(2,c (a+b x))}{e^3}+\frac {d^2 \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 )}{e^3}+\frac {d^2 \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))}{e^3}-\frac {d^2 \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 )}{e^3}+\frac {d^2 \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 )}{e^3}-\frac {d^2 \operatorname {PolyLog}\left (3,\frac {b (d+e x)}{b d-a e}\right )}{e^3}-\frac {d^2 \operatorname {PolyLog}(3,1-c (a+b x))}{e^3}-\frac {d^2 \operatorname {PolyLog}\left (3,-\frac {e (1-c (a+b x))}{b c (d+e x)}\right )}{e^3}+\frac {d^2 \operatorname {PolyLog}\left (3,\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{e^3}\right )+\frac {1}{2} b \left (\frac {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 ) a^2}{2 b^3}-\frac {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 a^2}{2 b^3}+\frac {h n \log (c (a+b x)) \log (d+e x) \log (1-c (a+b x)) a^2}{b^3}-\frac {g \operatorname {PolyLog}(2,c (a+b x)) a^2}{b^3}+\frac {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}{b^3}+\frac {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}{b^3}+\frac {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}{b^3}-\frac {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}{b^3}+\frac {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}{b^3}-\frac {h n \operatorname {PolyLog}\left (3,\frac {b (d+e x)}{b d-a e}\right ) a^2}{b^3}-\frac {h n \operatorname {PolyLog}(3,1-c (a+b x)) a^2}{b^3}-\frac {h n \operatorname {PolyLog}\left (3,-\frac {e (1-c (a+b x))}{b c (d+e x)}\right ) a^2}{b^3}+\frac {h n \operatorname {PolyLog}\left (3,\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right ) a^2}{b^3}+\frac {g x a}{b^2}-\frac {2 h n x a}{b^2}-\frac {h n (-a c-b x c+1) \log (-a c-b x c+1) a}{b^3 c}-\frac {d 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}{b^2 e}+\frac {h (d+e x) \log \left (f (d+e x)^n\right ) a}{b^2 e}-\frac {x \log (-a c-b x c+1) \left (g+h \log \left (f (d+e x)^n\right )\right ) a}{b^2}+\frac {(1-a 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}{b^3 c}-\frac {d h n \operatorname {PolyLog}\left (2,\frac {e (-a c-b x c+1)}{b c d-a c e+e}\right ) a}{b^2 e}+\frac {(1-a c) h n \operatorname {PolyLog}\left (2,\frac {b c (d+e x)}{b c d-a c e+e}\right ) a}{b^3 c}+\frac {h n x^2}{4 b}-\frac {(1-a c) g x}{2 b^2 c}+\frac {3 (1-a c) h n x}{4 b^2 c}-\frac {3 d h n x}{4 b e}-\frac {h n x^2 \log (-a c-b x c+1)}{4 b}+\frac {(1-a c)^2 h n \log (-a c-b x c+1)}{4 b^3 c^2}-\frac {d h n (-a c-b x c+1) \log (-a c-b x c+1)}{2 b^2 c e}+\frac {d^2 h n \log (d+e x)}{4 b e^2}-\frac {d^2 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 )}{2 b e^2}-\frac {(1-a c) h (d+e x) \log \left (f (d+e x)^n\right )}{2 b^2 c e}-\frac {x^2 \left (g+h \log \left (f (d+e x)^n\right )\right )}{4 b}+\frac {x^2 \log (-a c-b x c+1) \left (g+h \log \left (f (d+e x)^n\right )\right )}{2 b}-\frac {(1-a c)^2 \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 )}{2 b^3 c^2}-\frac {d^2 h n \operatorname {PolyLog}\left (2,\frac {e (-a c-b x c+1)}{b c d-a c e+e}\right )}{2 b e^2}-\frac {(1-a c)^2 h n \operatorname {PolyLog}\left (2,\frac {b c (d+e x)}{b c d-a c e+e}\right )}{2 b^3 c^2}\right )\)

input

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

output

(x^2*(g + h*Log[f*(d + e*x)^n])*PolyLog[2, c*(a + b*x)])/2 - (e*h*n*((d*x) 
/e^2 + (a*x)/(2*b*e) - ((1 - a*c)*x)/(4*b*c*e) - x^2/(8*e) - ((1 - a*c)^2* 
Log[1 - a*c - b*c*x])/(4*b^2*c^2*e) + (x^2*Log[1 - a*c - b*c*x])/(4*e) + ( 
d*(1 - a*c - b*c*x)*Log[1 - a*c - b*c*x])/(b*c*e^2) + (a*(1 - a*c - b*c*x) 
*Log[1 - a*c - b*c*x])/(2*b^2*c*e) + (d^2*(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*e^3) + ( 
d^2*Log[c*(a + b*x)]*Log[d + e*x]*Log[1 - c*(a + b*x)])/e^3 - (d^2*(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*e^3) - (a*d*PolyLo 
g[2, c*(a + b*x)])/(b*e^2) - (a^2*PolyLog[2, c*(a + b*x)])/(2*b^2*e) - (d* 
x*PolyLog[2, c*(a + b*x)])/e^2 + (x^2*PolyLog[2, c*(a + b*x)])/(2*e) + (d^ 
2*Log[d + e*x]*PolyLog[2, c*(a + b*x)])/e^3 + (d^2*(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)])/e^3 + (d^2*(Log[d + e*x] - Log[(b*(d + e*x))/((b*d - a*e 
)*(1 - c*(a + b*x)))])*PolyLog[2, 1 - c*(a + b*x)])/e^3 - (d^2*Log[(b*(d + 
 e*x))/((b*d - a*e)*(1 - c*(a + b*x)))]*PolyLog[2, -((e*(1 - c*(a + b*x))) 
/(b*c*(d + e*x)))])/e^3 + (d^2*Log[(b*(d + e*x))/((b*d - a*e)*(1 - c*(a + 
b*x)))]*PolyLog[2, ((b*d - a*e)*(1 - c*(a + b*x)))/(b*(d + e*x))])/e^3 - ( 
d^2*PolyLog[3, (b*(d + e*x))/(b*d - a*e)])/e^3 - (d^2*PolyLog[3, 1 - c*...
 

3.2.78.3.1 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]
 

3.2.78.4 Maple [F]

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

input

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

output

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

3.2.78.5 Fricas [F]

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

input

integrate(x*(g+h*log(f*(e*x+d)^n))*polylog(2,c*(b*x+a)),x, algorithm="fric 
as")
 

output

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

3.2.78.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.2.78.7 Maxima [F]

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

input

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

output

1/4*(2*e^2*h*x^2*log((e*x + d)^n) + 2*d*e*h*n*x - 2*d^2*h*n*log(e*x + d) - 
 (e^2*h*n - 2*e^2*h*log(f) - 2*e^2*g)*x^2)*dilog(b*c*x + a*c)/e^2 + integr 
ate(1/4*(2*b*e^2*h*x^2*log(-b*c*x - a*c + 1)*log((e*x + d)^n) + (2*b*d*e*h 
*n*x - 2*b*d^2*h*n*log(e*x + d) - (b*e^2*h*n - 2*b*e^2*h*log(f) - 2*b*e^2* 
g)*x^2)*log(-b*c*x - a*c + 1))/(b*e^2*x + a*e^2), x)
 

3.2.78.8 Giac [F]

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

input

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

output

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

3.2.78.9 Mupad [F(-1)]

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

input

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

output

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