Optimal. Leaf size=65 \[ \frac{x \text{PolyLog}\left (n+1,d \left (F^{c (a+b x)}\right )^p\right )}{b c p \log (F)}-\frac{\text{PolyLog}\left (n+2,d \left (F^{c (a+b x)}\right )^p\right )}{b^2 c^2 p^2 \log ^2(F)} \]
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Rubi [A] time = 0.0331011, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {6609, 2282, 6589} \[ \frac{x \text{PolyLog}\left (n+1,d \left (F^{c (a+b x)}\right )^p\right )}{b c p \log (F)}-\frac{\text{PolyLog}\left (n+2,d \left (F^{c (a+b x)}\right )^p\right )}{b^2 c^2 p^2 \log ^2(F)} \]
Antiderivative was successfully verified.
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Rule 6609
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int x \text{Li}_n\left (d \left (F^{c (a+b x)}\right )^p\right ) \, dx &=\frac{x \text{Li}_{1+n}\left (d \left (F^{c (a+b x)}\right )^p\right )}{b c p \log (F)}-\frac{\int \text{Li}_{1+n}\left (d \left (F^{c (a+b x)}\right )^p\right ) \, dx}{b c p \log (F)}\\ &=\frac{x \text{Li}_{1+n}\left (d \left (F^{c (a+b x)}\right )^p\right )}{b c p \log (F)}-\frac{\operatorname{Subst}\left (\int \frac{\text{Li}_{1+n}\left (d x^p\right )}{x} \, dx,x,F^{c (a+b x)}\right )}{b^2 c^2 p \log ^2(F)}\\ &=\frac{x \text{Li}_{1+n}\left (d \left (F^{c (a+b x)}\right )^p\right )}{b c p \log (F)}-\frac{\text{Li}_{2+n}\left (d \left (F^{c (a+b x)}\right )^p\right )}{b^2 c^2 p^2 \log ^2(F)}\\ \end{align*}
Mathematica [A] time = 0.0052208, size = 65, normalized size = 1. \[ \frac{x \text{PolyLog}\left (n+1,d \left (F^{c (a+b x)}\right )^p\right )}{b c p \log (F)}-\frac{\text{PolyLog}\left (n+2,d \left (F^{c (a+b x)}\right )^p\right )}{b^2 c^2 p^2 \log ^2(F)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.055, size = 0, normalized size = 0. \begin{align*} \int x{\it polylog} \left ( n,d \left ({F}^{c \left ( bx+a \right ) } \right ) ^{p} \right ) \, dx \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 x{\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 (x{\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 x \operatorname{Li}_{n}\left (d \left (F^{a c} F^{b c x}\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 x{\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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