Optimal. Leaf size=78 \[ \frac{a (d x)^{m+2} \text{Hypergeometric2F1}(1,m+2,m+3,a x)}{d^2 (m+1)^2 (m+2)}+\frac{(d x)^{m+1} \text{PolyLog}(2,a x)}{d (m+1)}+\frac{\log (1-a x) (d x)^{m+1}}{d (m+1)^2} \]
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Rubi [A] time = 0.0487671, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {6591, 2395, 64} \[ \frac{(d x)^{m+1} \text{PolyLog}(2,a x)}{d (m+1)}+\frac{a (d x)^{m+2} \, _2F_1(1,m+2;m+3;a x)}{d^2 (m+1)^2 (m+2)}+\frac{\log (1-a x) (d x)^{m+1}}{d (m+1)^2} \]
Antiderivative was successfully verified.
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Rule 6591
Rule 2395
Rule 64
Rubi steps
\begin{align*} \int (d x)^m \text{Li}_2(a x) \, dx &=\frac{(d x)^{1+m} \text{Li}_2(a x)}{d (1+m)}+\frac{\int (d x)^m \log (1-a x) \, dx}{1+m}\\ &=\frac{(d x)^{1+m} \log (1-a x)}{d (1+m)^2}+\frac{(d x)^{1+m} \text{Li}_2(a x)}{d (1+m)}+\frac{a \int \frac{(d x)^{1+m}}{1-a x} \, dx}{d (1+m)^2}\\ &=\frac{a (d x)^{2+m} \, _2F_1(1,2+m;3+m;a x)}{d^2 (1+m)^2 (2+m)}+\frac{(d x)^{1+m} \log (1-a x)}{d (1+m)^2}+\frac{(d x)^{1+m} \text{Li}_2(a x)}{d (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0387445, size = 53, normalized size = 0.68 \[ \frac{x (d x)^m (a x \text{Hypergeometric2F1}(1,m+2,m+3,a x)+(m+2) ((m+1) \text{PolyLog}(2,a x)+\log (1-a x)))}{(m+1)^2 (m+2)} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.227, size = 144, normalized size = 1.9 \begin{align*}{\frac{ \left ( dx \right ) ^{m}{x}^{-m} \left ( -a \right ) ^{-m}}{a} \left ({\frac{{x}^{m} \left ( -a \right ) ^{m} \left ( -a{m}^{2}x-2\,amx-{m}^{2}-3\,m-2 \right ) }{ \left ( 1+m \right ) ^{3} \left ( 2+m \right ) m}}-{\frac{{x}^{1+m}a \left ( -a \right ) ^{m} \left ( -m-2 \right ) \ln \left ( -ax+1 \right ) }{ \left ( 1+m \right ) ^{2} \left ( 2+m \right ) }}+{\frac{{x}^{1+m}a \left ( -a \right ) ^{m}{\it polylog} \left ( 2,ax \right ) }{1+m}}+{\frac{{x}^{m} \left ( -a \right ) ^{m}{\it LerchPhi} \left ( ax,1,m \right ) }{ \left ( 1+m \right ) ^{2}}} \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*} -a d^{m} \int -\frac{x x^{m}}{m^{2} -{\left (a m^{2} + 2 \, a m + a\right )} x + 2 \, m + 1}\,{d x} + \frac{{\left (d^{m} m + d^{m}\right )} x x^{m}{\rm Li}_2\left (a x\right ) + d^{m} x x^{m} \log \left (-a x + 1\right )}{m^{2} + 2 \, m + 1} \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 (\left (d x\right )^{m}{\rm Li}_2\left (a x\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 \left (d x\right )^{m} \operatorname{Li}_{2}\left (a x\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 \left (d x\right )^{m}{\rm Li}_2\left (a x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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