Optimal. Leaf size=100 \[ -\frac{2 x \text{PolyLog}\left (n+2,d \left (F^{c (a+b x)}\right )^p\right )}{b^2 c^2 p^2 \log ^2(F)}+\frac{2 \text{PolyLog}\left (n+3,d \left (F^{c (a+b x)}\right )^p\right )}{b^3 c^3 p^3 \log ^3(F)}+\frac{x^2 \text{PolyLog}\left (n+1,d \left (F^{c (a+b x)}\right )^p\right )}{b c p \log (F)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0579408, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {6609, 2282, 6589} \[ -\frac{2 x \text{PolyLog}\left (n+2,d \left (F^{c (a+b x)}\right )^p\right )}{b^2 c^2 p^2 \log ^2(F)}+\frac{2 \text{PolyLog}\left (n+3,d \left (F^{c (a+b x)}\right )^p\right )}{b^3 c^3 p^3 \log ^3(F)}+\frac{x^2 \text{PolyLog}\left (n+1,d \left (F^{c (a+b x)}\right )^p\right )}{b c p \log (F)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6609
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int x^2 \text{Li}_n\left (d \left (F^{c (a+b x)}\right )^p\right ) \, dx &=\frac{x^2 \text{Li}_{1+n}\left (d \left (F^{c (a+b x)}\right )^p\right )}{b c p \log (F)}-\frac{2 \int x \text{Li}_{1+n}\left (d \left (F^{c (a+b x)}\right )^p\right ) \, dx}{b c p \log (F)}\\ &=\frac{x^2 \text{Li}_{1+n}\left (d \left (F^{c (a+b x)}\right )^p\right )}{b c p \log (F)}-\frac{2 x \text{Li}_{2+n}\left (d \left (F^{c (a+b x)}\right )^p\right )}{b^2 c^2 p^2 \log ^2(F)}+\frac{2 \int \text{Li}_{2+n}\left (d \left (F^{c (a+b x)}\right )^p\right ) \, dx}{b^2 c^2 p^2 \log ^2(F)}\\ &=\frac{x^2 \text{Li}_{1+n}\left (d \left (F^{c (a+b x)}\right )^p\right )}{b c p \log (F)}-\frac{2 x \text{Li}_{2+n}\left (d \left (F^{c (a+b x)}\right )^p\right )}{b^2 c^2 p^2 \log ^2(F)}+\frac{2 \operatorname{Subst}\left (\int \frac{\text{Li}_{2+n}\left (d x^p\right )}{x} \, dx,x,F^{c (a+b x)}\right )}{b^3 c^3 p^2 \log ^3(F)}\\ &=\frac{x^2 \text{Li}_{1+n}\left (d \left (F^{c (a+b x)}\right )^p\right )}{b c p \log (F)}-\frac{2 x \text{Li}_{2+n}\left (d \left (F^{c (a+b x)}\right )^p\right )}{b^2 c^2 p^2 \log ^2(F)}+\frac{2 \text{Li}_{3+n}\left (d \left (F^{c (a+b x)}\right )^p\right )}{b^3 c^3 p^3 \log ^3(F)}\\ \end{align*}
Mathematica [A] time = 0.0058507, size = 100, normalized size = 1. \[ -\frac{2 x \text{PolyLog}\left (n+2,d \left (F^{c (a+b x)}\right )^p\right )}{b^2 c^2 p^2 \log ^2(F)}+\frac{2 \text{PolyLog}\left (n+3,d \left (F^{c (a+b x)}\right )^p\right )}{b^3 c^3 p^3 \log ^3(F)}+\frac{x^2 \text{PolyLog}\left (n+1,d \left (F^{c (a+b x)}\right )^p\right )}{b c p \log (F)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.053, size = 0, normalized size = 0. \begin{align*} \int{x}^{2}{\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.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2}{\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.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{2}{\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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2}{\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.
[In]
[Out]