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

Optimal. Leaf size=35 \[ \frac{\text{PolyLog}\left (n+1,e \left (\frac{a+b x}{c+d x}\right )^n\right )}{n (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.071321, antiderivative size = 35, 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} \[ \frac{\text{PolyLog}\left (n+1,e \left (\frac{a+b x}{c+d x}\right )^n\right )}{n (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

PolyLog[1 + n, 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}_n\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)} \, dx &=\frac{\text{Li}_{1+n}\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(b c-a d) n}\\ \end{align*}

Mathematica [A]  time = 0.0198299, size = 34, normalized size = 0.97 \[ \frac{\text{PolyLog}\left (n+1,e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b c n-a d n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.891, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( bx+a \right ) \left ( dx+c \right ) }{\it polylog} \left ( n,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(n,e*((b*x+a)/(d*x+c))^n)/(b*x+a)/(d*x+c),x)

[Out]

int(polylog(n,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*} \int \frac{{\rm Li}_{n}(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(n,e*((b*x+a)/(d*x+c))^n)/(b*x+a)/(d*x+c),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\rm polylog}\left (n, e \left (\frac{b x + a}{d x + c}\right )^{n}\right )}{b d x^{2} + a c +{\left (b c + a d\right )} x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(polylog(n, e*((b*x + a)/(d*x + c))^n)/(b*d*x^2 + a*c + (b*c + a*d)*x), x)

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

[Out]

Integral(polylog(n, 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}_{n}(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(n,e*((b*x+a)/(d*x+c))^n)/(b*x+a)/(d*x+c),x, algorithm="giac")

[Out]

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