Optimal. Leaf size=31 \[ \frac{\text{PolyLog}\left (n+1,d \left (F^{c (a+b x)}\right )^p\right )}{b c p \log (F)} \]
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Rubi [A] time = 0.0177012, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2282, 6589} \[ \frac{\text{PolyLog}\left (n+1,d \left (F^{c (a+b x)}\right )^p\right )}{b c p \log (F)} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \text{Li}_n\left (d \left (F^{c (a+b x)}\right )^p\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\text{Li}_n\left (d x^p\right )}{x} \, dx,x,F^{c (a+b x)}\right )}{b c \log (F)}\\ &=\frac{\text{Li}_{1+n}\left (d \left (F^{c (a+b x)}\right )^p\right )}{b c p \log (F)}\\ \end{align*}
Mathematica [A] time = 0.004743, size = 31, normalized size = 1. \[ \frac{\text{PolyLog}\left (n+1,d \left (F^{c (a+b x)}\right )^p\right )}{b c p \log (F)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.061, size = 32, normalized size = 1. \begin{align*}{\frac{{\it polylog} \left ( 1+n,d \left ({F}^{c \left ( bx+a \right ) } \right ) ^{p} \right ) }{pbc\ln \left ( F \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\rm Li}_{n}({\left (F^{{\left (b x + a\right )} c}\right )}^{p} d)\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\rm polylog}\left (n,{\left (F^{b c x + a c}\right )}^{p} d\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{Li}_{n}\left (d \left (F^{c \left (a + b x\right )}\right )^{p}\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\rm Li}_{n}({\left (F^{{\left (b x + a\right )} c}\right )}^{p} d)\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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