\(\int \frac {\operatorname {PolyLog}(2,c (a+b x))}{(d+e x)^3} \, dx\) [143]

Optimal result

Integrand size = 17, antiderivative size = 278 \[ \int \frac {\operatorname {PolyLog}(2,c (a+b x))}{(d+e x)^3} \, dx=\frac {b^2 c \log (1-a c-b c x)}{2 e (b d-a e) (b c d+e-a c e)}-\frac {b \log (1-a c-b c x)}{2 e (b d-a e) (d+e x)}-\frac {b^2 c \log (d+e x)}{2 e (b d-a e) (b c d+e-a c e)}+\frac {b^2 \log (1-a c-b c x) \log \left (\frac {b c (d+e x)}{b c d+e-a c e}\right )}{2 e (b d-a e)^2}+\frac {b^2 \operatorname {PolyLog}(2,c (a+b x))}{2 e (b d-a e)^2}-\frac {\operatorname {PolyLog}(2,c (a+b x))}{2 e (d+e x)^2}+\frac {b^2 \operatorname {PolyLog}\left (2,\frac {e (1-a c-b c x)}{b c d+e-a c e}\right )}{2 e (b d-a e)^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.68 \[ \int \frac {\operatorname {PolyLog}(2,c (a+b x))}{(d+e x)^3} \, dx=\frac {-\frac {\operatorname {PolyLog}(2,c (a+b x))}{(d+e x)^2}+\frac {b \left (-\frac {(b d-a e) \log (1-a c-b c x)}{d+e x}+\frac {b c (b d-a e) (\log (1-a c-b c x)-\log (d+e x))}{b c d+e-a c e}+b \log (1-a c-b c x) \log \left (\frac {b c (d+e x)}{b c d+e-a c e}\right )+b \operatorname {PolyLog}(2,c (a+b x))+b \operatorname {PolyLog}\left (2,\frac {e (-1+a c+b c x)}{-b c d+(-1+a c) e}\right )\right )}{(b d-a e)^2}}{2 e} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.88, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {7152, 2865, 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.

\(\displaystyle \int \frac {\operatorname {PolyLog}(2,c (a+b x))}{(d+e x)^3} \, dx\)

\(\Big \downarrow \) 7152

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

\(\Big \downarrow \) 2865

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

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

rule 2865

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Sy 
mbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, 
Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunctionQ[ 
RFx, x] && IntegerQ[p]
 

rule 7152

Int[((d_.) + (e_.)*(x_))^(m_.)*PolyLog[2, (c_.)*((a_.) + (b_.)*(x_))], x_Sy 
mbol] :> Simp[(d + e*x)^(m + 1)*(PolyLog[2, c*(a + b*x)]/(e*(m + 1))), x] + 
 Simp[b/(e*(m + 1))   Int[(d + e*x)^(m + 1)*(Log[1 - a*c - b*c*x]/(a + b*x) 
), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[m, -1]
 

Maple [A] (verified)

Time = 10.60 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.13

method	result	size
parts	\(-\frac {\operatorname {polylog}\left (2, c \left (b x +a \right )\right )}{2 e \left (e x +d \right )^{2}}+\frac {\frac {c \,b^{2} e \left (\frac {\operatorname {dilog}\left (\frac {a e c -b c d +e \left (-b c x -a c +1\right )-e}{a e c -b c d -e}\right )}{e}+\frac {\ln \left (-b c x -a c +1\right ) \ln \left (\frac {a e c -b c d +e \left (-b c x -a c +1\right )-e}{a e c -b c d -e}\right )}{e}\right )}{\left (a e -b d \right )^{2}}+\frac {c \,b^{2} \operatorname {dilog}\left (-b c x -a c +1\right )}{\left (a e -b d \right )^{2}}+\frac {c^{2} b^{2} e \left (-\frac {\ln \left (a e c -b c d +e \left (-b c x -a c +1\right )-e \right )}{\left (a e c -b c d -e \right ) e}+\frac {\ln \left (-b c x -a c +1\right ) \left (-b c x -a c +1\right )}{\left (a e c -b c d -e \right ) \left (a e c -b c d +e \left (-b c x -a c +1\right )-e \right )}\right )}{a e -b d}}{2 e c}\)	\(315\)
derivativedivides	\(\frac {-\frac {c^{3} b^{3} \operatorname {polylog}\left (2, b c x +a c \right )}{2 \left (a e c -b c d -e \left (b c x +a c \right )\right )^{2} e}-\frac {c^{3} b^{3} \left (-\frac {\operatorname {dilog}\left (-b c x -a c +1\right )}{c^{2} \left (a e -b d \right )^{2}}-\frac {e \left (-\frac {\ln \left (a e c -b c d +e \left (-b c x -a c +1\right )-e \right )}{\left (a e c -b c d -e \right ) e}+\frac {\ln \left (-b c x -a c +1\right ) \left (-b c x -a c +1\right )}{\left (a e c -b c d -e \right ) \left (a e c -b c d +e \left (-b c x -a c +1\right )-e \right )}\right )}{c \left (a e -b d \right )}-\frac {e \left (\frac {\operatorname {dilog}\left (\frac {a e c -b c d +e \left (-b c x -a c +1\right )-e}{a e c -b c d -e}\right )}{e}+\frac {\ln \left (-b c x -a c +1\right ) \ln \left (\frac {a e c -b c d +e \left (-b c x -a c +1\right )-e}{a e c -b c d -e}\right )}{e}\right )}{c^{2} \left (a e -b d \right )^{2}}\right )}{2 e}}{c b}\)	\(346\)
default	\(\frac {-\frac {c^{3} b^{3} \operatorname {polylog}\left (2, b c x +a c \right )}{2 \left (a e c -b c d -e \left (b c x +a c \right )\right )^{2} e}-\frac {c^{3} b^{3} \left (-\frac {\operatorname {dilog}\left (-b c x -a c +1\right )}{c^{2} \left (a e -b d \right )^{2}}-\frac {e \left (-\frac {\ln \left (a e c -b c d +e \left (-b c x -a c +1\right )-e \right )}{\left (a e c -b c d -e \right ) e}+\frac {\ln \left (-b c x -a c +1\right ) \left (-b c x -a c +1\right )}{\left (a e c -b c d -e \right ) \left (a e c -b c d +e \left (-b c x -a c +1\right )-e \right )}\right )}{c \left (a e -b d \right )}-\frac {e \left (\frac {\operatorname {dilog}\left (\frac {a e c -b c d +e \left (-b c x -a c +1\right )-e}{a e c -b c d -e}\right )}{e}+\frac {\ln \left (-b c x -a c +1\right ) \ln \left (\frac {a e c -b c d +e \left (-b c x -a c +1\right )-e}{a e c -b c d -e}\right )}{e}\right )}{c^{2} \left (a e -b d \right )^{2}}\right )}{2 e}}{c b}\)	\(346\)

Input:

int(polylog(2,c*(b*x+a))/(e*x+d)^3,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

integrate(polylog(2,c*(b*x+a))/(e*x+d)**3,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.36 \[ \int \frac {\operatorname {PolyLog}(2,c (a+b x))}{(d+e x)^3} \, dx=\frac {b^{2} c \log \left (b c x + a c - 1\right )}{2 \, {\left (b^{2} c d^{2} e - {\left (2 \, a b c - b\right )} d e^{2} + {\left (a^{2} c - a\right )} e^{3}\right )}} - \frac {b^{2} c \log \left (e x + d\right )}{2 \, {\left (b^{2} c d^{2} e - {\left (2 \, a b c - b\right )} d e^{2} + {\left (a^{2} c - a\right )} e^{3}\right )}} - \frac {{\left (\log \left (b c x + a c\right ) \log \left (-b c x - a c + 1\right ) + {\rm Li}_2\left (-b c x - a c + 1\right )\right )} b^{2}}{2 \, {\left (b^{2} d^{2} e - 2 \, a b d e^{2} + a^{2} e^{3}\right )}} + \frac {{\left (\log \left (-b c x - a c + 1\right ) \log \left (\frac {b c e x + {\left (a c - 1\right )} e}{b c d - {\left (a c - 1\right )} e} + 1\right ) + {\rm Li}_2\left (-\frac {b c e x + {\left (a c - 1\right )} e}{b c d - {\left (a c - 1\right )} e}\right )\right )} b^{2}}{2 \, {\left (b^{2} d^{2} e - 2 \, a b d e^{2} + a^{2} e^{3}\right )}} - \frac {{\left (b d - a e\right )} {\rm Li}_2\left (b c x + a c\right ) + {\left (b e x + b d\right )} \log \left (-b c x - a c + 1\right )}{2 \, {\left (b d^{3} e - a d^{2} e^{2} + {\left (b d e^{3} - a e^{4}\right )} x^{2} + 2 \, {\left (b d^{2} e^{2} - a d e^{3}\right )} x\right )}} \] Input:

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

Output:

1/2*b^2*c*log(b*c*x + a*c - 1)/(b^2*c*d^2*e - (2*a*b*c - b)*d*e^2 + (a^2*c 
 - a)*e^3) - 1/2*b^2*c*log(e*x + d)/(b^2*c*d^2*e - (2*a*b*c - b)*d*e^2 + ( 
a^2*c - a)*e^3) - 1/2*(log(b*c*x + a*c)*log(-b*c*x - a*c + 1) + dilog(-b*c 
*x - a*c + 1))*b^2/(b^2*d^2*e - 2*a*b*d*e^2 + a^2*e^3) + 1/2*(log(-b*c*x - 
 a*c + 1)*log((b*c*e*x + (a*c - 1)*e)/(b*c*d - (a*c - 1)*e) + 1) + dilog(- 
(b*c*e*x + (a*c - 1)*e)/(b*c*d - (a*c - 1)*e)))*b^2/(b^2*d^2*e - 2*a*b*d*e 
^2 + a^2*e^3) - 1/2*((b*d - a*e)*dilog(b*c*x + a*c) + (b*e*x + b*d)*log(-b 
*c*x - a*c + 1))/(b*d^3*e - a*d^2*e^2 + (b*d*e^3 - a*e^4)*x^2 + 2*(b*d^2*e 
^2 - a*d*e^3)*x)
 

Giac [F]

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

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

Output:

integrate(dilog((b*x + a)*c)/(e*x + d)^3, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(polylog(2,a*c + b*c*x)/(d**3 + 3*d**2*e*x + 3*d*e**2*x**2 + e**3*x**3) 
,x)