Optimal. Leaf size=35 \[ \frac {\text {PolyLog}\left (1+n,e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(b c-a d) n} \]
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Rubi [A]
time = 0.04, antiderivative size = 35, normalized size of antiderivative = 1.00, 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 = {6745}
\begin {gather*} \frac {\text {Li}_{n+1}\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{n (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 6745
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*}
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Mathematica [A]
time = 0.02, size = 34, normalized size = 0.97 \begin {gather*} \frac {\text {PolyLog}\left (1+n,e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b c n-a d n} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.33, size = 0, normalized size = 0.00 \[\int \frac {\polylog \left (n , e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}{\left (b x +a \right ) \left (d x +c \right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\mathrm {polylog}\left (n,e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}{\left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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