3.151 \(\int \frac{\text{PolyLog}(2,e (\frac{a+b x}{c+d x})^n)}{(a+b x) (c+d x)} \, dx\)

Optimal. Leaf size=33 \[ \frac{\text{PolyLog}\left (3,e \left (\frac{a+b x}{c+d x}\right )^n\right )}{n (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.0612712, 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} \[ \frac{\text{PolyLog}\left (3,e \left (\frac{a+b x}{c+d x}\right )^n\right )}{n (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

PolyLog[3, 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}_2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)} \, dx &=\frac{\text{Li}_3\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(b c-a d) n}\\ \end{align*}

Mathematica [A]  time = 0.006995, size = 32, normalized size = 0.97 \[ \frac{\text{PolyLog}\left (3,e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b c n-a d n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.805, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( bx+a \right ) \left ( dx+c \right ) }{\it polylog} \left ( 2,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(2,e*((b*x+a)/(d*x+c))^n)/(b*x+a)/(d*x+c),x)

[Out]

int(polylog(2,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{2 \,{\left (\log \left (b x + a\right ) - \log \left (d x + c\right )\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 \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 )} \log \left (-{\left (b x + a\right )}^{n} e +{\left (d x + c\right )}^{n}\right ) -{\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 )} \log \left ({\left (d x + c\right )}^{n}\right )}{2 \,{\left (b c - a d\right )}} + \int -\frac{{\left (e n^{2} \log \left (b x + a\right )^{2} - 2 \, e n^{2} \log \left (b x + a\right ) \log \left (d x + c\right ) + e n^{2} \log \left (d x + c\right )^{2}\right )}{\left (b x + a\right )}^{n}}{2 \,{\left ({\left (b d e x^{2} + a c e +{\left (b c + a d\right )} e 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(2,e*((b*x+a)/(d*x+c))^n)/(b*x+a)/(d*x+c),x, algorithm="maxima")

[Out]

1/2*(2*(log(b*x + a) - log(d*x + c))*dilog(e*e^(n*log(b*x + a) - n*log(d*x + c))) + (n*log(b*x + a)^2 - 2*n*lo
g(b*x + a)*log(d*x + c) + n*log(d*x + c)^2)*log(-(b*x + a)^n*e + (d*x + c)^n) - (n*log(b*x + a)^2 - 2*n*log(b*
x + a)*log(d*x + c) + n*log(d*x + c)^2)*log((d*x + c)^n))/(b*c - a*d) + integrate(-1/2*(e*n^2*log(b*x + a)^2 -
2*e*n^2*log(b*x + a)*log(d*x + c) + e*n^2*log(d*x + c)^2)*(b*x + a)^n/((b*d*e*x^2 + a*c*e + (b*c + 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 [A]  time = 2.7098, size = 74, normalized size = 2.24 \begin{align*} \frac{{\rm polylog}\left (3, e \left (\frac{b x + a}{d x + c}\right )^{n}\right )}{{\left (b c - a d\right )} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Li}_2\left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )}{{\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(2,e*((b*x+a)/(d*x+c))^n)/(b*x+a)/(d*x+c),x, algorithm="giac")

[Out]

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