3.150 \(\int \frac{\text{PolyLog}(3,e (\frac{a+b x}{c+d x})^n)}{(a+b x) (c+d x)} \, dx\)

Optimal. Leaf size=33 \[ \frac{\text{PolyLog}\left (4,e \left (\frac{a+b x}{c+d x}\right )^n\right )}{n (b c-a d)} \]

[Out]

PolyLog[4, e*((a + b*x)/(c + d*x))^n]/((b*c - a*d)*n)

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Rubi [A]  time = 0.0611184, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.029, Rules used = {6610} \[ \frac{\text{PolyLog}\left (4,e \left (\frac{a+b x}{c+d x}\right )^n\right )}{n (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

PolyLog[4, e*((a + b*x)/(c + d*x))^n]/((b*c - a*d)*n)

Rule 6610

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

Rubi steps

\begin{align*} \int \frac{\text{Li}_3\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)} \, dx &=\frac{\text{Li}_4\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(b c-a d) n}\\ \end{align*}

Mathematica [A]  time = 0.007177, size = 32, normalized size = 0.97 \[ \frac{\text{PolyLog}\left (4,e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b c n-a d n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

PolyLog[4, e*((a + b*x)/(c + d*x))^n]/(b*c*n - a*d*n)

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Maple [F]  time = 0.703, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( bx+a \right ) \left ( dx+c \right ) }{\it polylog} \left ( 3,e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{3 \,{\left (n \log \left (b x + a\right )^{2} - 2 \, n \log \left (b x + a\right ) \log \left (d x + c\right ) + n \log \left (d x + c\right )^{2}\right )}{\rm Li}_2\left (e e^{\left (n \log \left (b x + a\right ) - n \log \left (d x + c\right )\right )}\right ) +{\left (n^{2} \log \left (b x + a\right )^{3} - 3 \, n^{2} \log \left (b x + a\right )^{2} \log \left (d x + c\right ) + 3 \, n^{2} \log \left (b x + a\right ) \log \left (d x + c\right )^{2} - n^{2} \log \left (d x + c\right )^{3}\right )} \log \left (-{\left (b x + a\right )}^{n} e +{\left (d x + c\right )}^{n}\right ) -{\left (n^{2} \log \left (b x + a\right )^{3} - 3 \, n^{2} \log \left (b x + a\right )^{2} \log \left (d x + c\right ) + 3 \, n^{2} \log \left (b x + a\right ) \log \left (d x + c\right )^{2} - n^{2} \log \left (d x + c\right )^{3}\right )} \log \left ({\left (d x + c\right )}^{n}\right ) - 6 \,{\left (\log \left (b x + a\right ) - \log \left (d x + c\right )\right )}{\rm Li}_{3}(e e^{\left (n \log \left (b x + a\right ) - n \log \left (d x + c\right )\right )})}{6 \,{\left (b c - a d\right )}} + \int \frac{{\left (e n^{3} \log \left (b x + a\right )^{3} - 3 \, e n^{3} \log \left (b x + a\right )^{2} \log \left (d x + c\right ) + 3 \, e n^{3} \log \left (b x + a\right ) \log \left (d x + c\right )^{2} - e n^{3} \log \left (d x + c\right )^{3}\right )}{\left (b x + a\right )}^{n}}{6 \,{\left ({\left (b d e x^{2} + a c e +{\left (b c e + a d e\right )} x\right )}{\left (b x + a\right )}^{n} -{\left (b d x^{2} + a c +{\left (b c + a d\right )} x\right )}{\left (d x + c\right )}^{n}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(3*(n*log(b*x + a)^2 - 2*n*log(b*x + a)*log(d*x + c) + n*log(d*x + c)^2)*dilog(e*e^(n*log(b*x + a) - n*lo
g(d*x + c))) + (n^2*log(b*x + a)^3 - 3*n^2*log(b*x + a)^2*log(d*x + c) + 3*n^2*log(b*x + a)*log(d*x + c)^2 - n
^2*log(d*x + c)^3)*log(-(b*x + a)^n*e + (d*x + c)^n) - (n^2*log(b*x + a)^3 - 3*n^2*log(b*x + a)^2*log(d*x + c)
+ 3*n^2*log(b*x + a)*log(d*x + c)^2 - n^2*log(d*x + c)^3)*log((d*x + c)^n) - 6*(log(b*x + a) - log(d*x + c))*
polylog(3, e*e^(n*log(b*x + a) - n*log(d*x + c))))/(b*c - a*d) + integrate(1/6*(e*n^3*log(b*x + a)^3 - 3*e*n^3
*log(b*x + a)^2*log(d*x + c) + 3*e*n^3*log(b*x + a)*log(d*x + c)^2 - e*n^3*log(d*x + c)^3)*(b*x + a)^n/((b*d*e
*x^2 + a*c*e + (b*c*e + a*d*e)*x)*(b*x + a)^n - (b*d*x^2 + a*c + (b*c + a*d)*x)*(d*x + c)^n), x)

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Fricas [C]  time = 3.45558, size = 782, normalized size = 23.7 \begin{align*} -\frac{n^{2}{\rm \%iint}\left (a, b, c, d, e, n, x, -\frac{n \log \left (-e \left (\frac{b x + a}{d x + c}\right )^{n} + 1\right )}{b x + a}, -\frac{n x \log \left (-e \left (\frac{b x + a}{d x + c}\right )^{n} + 1\right )}{b x + a}, \frac{n \log \left (-e \left (\frac{b x + a}{d x + c}\right )^{n} + 1\right )}{d x + c}, \frac{n x \log \left (-e \left (\frac{b x + a}{d x + c}\right )^{n} + 1\right )}{d x + c}, -\frac{\log \left (-e \left (\frac{b x + a}{d x + c}\right )^{n} + 1\right )}{e}, -\log \left (-e \left (\frac{b x + a}{d x + c}\right )^{n} + 1\right ) \log \left (\frac{b x + a}{d x + c}\right ), -\frac{{\left (b c - a d\right )} n \log \left (-e \left (\frac{b x + a}{d x + c}\right )^{n} + 1\right )}{b d x^{2} + a c +{\left (b c + a d\right )} x}\right ) \log \left (\frac{b x + a}{d x + c}\right )^{2} - n^{2}{\rm Li}_2\left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) \log \left (\frac{b x + a}{d x + c}\right )^{2} - 2 \,{\rm polylog}\left (4, e \left (\frac{b x + a}{d x + c}\right )^{n}\right )}{2 \,{\left (b c - a d\right )} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(n^2*\%iint(a, b, c, d, e, n, x, -n*log(-e*((b*x + a)/(d*x + c))^n + 1)/(b*x + a), -n*x*log(-e*((b*x + a)/
(d*x + c))^n + 1)/(b*x + a), n*log(-e*((b*x + a)/(d*x + c))^n + 1)/(d*x + c), n*x*log(-e*((b*x + a)/(d*x + c))
^n + 1)/(d*x + c), -log(-e*((b*x + a)/(d*x + c))^n + 1)/e, -log(-e*((b*x + a)/(d*x + c))^n + 1)*log((b*x + a)/
(d*x + c)), -(b*c - a*d)*n*log(-e*((b*x + a)/(d*x + c))^n + 1)/(b*d*x^2 + a*c + (b*c + a*d)*x))*log((b*x + a)/
(d*x + c))^2 - n^2*dilog(e*((b*x + a)/(d*x + c))^n)*log((b*x + a)/(d*x + c))^2 - 2*polylog(4, e*((b*x + a)/(d*
x + c))^n))/((b*c - a*d)*n)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{Li}_{3}\left (e \left (\frac{a}{c + d x} + \frac{b x}{c + d x}\right )^{n}\right )}{\left (a + b x\right ) \left (c + d x\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Li}_{3}(e \left (\frac{b x + a}{d x + c}\right )^{n})}{{\left (b x + a\right )}{\left (d x + c\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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