Optimal. Leaf size=86 \[ \frac{1}{5} x^5 \text{PolyLog}(2,a x)-\frac{x^3}{75 a^2}-\frac{x^2}{50 a^3}-\frac{x}{25 a^4}-\frac{\log (1-a x)}{25 a^5}-\frac{x^4}{100 a}+\frac{1}{25} x^5 \log (1-a x)-\frac{x^5}{125} \]
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Rubi [A] time = 0.0533019, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6591, 2395, 43} \[ \frac{1}{5} x^5 \text{PolyLog}(2,a x)-\frac{x^3}{75 a^2}-\frac{x^2}{50 a^3}-\frac{x}{25 a^4}-\frac{\log (1-a x)}{25 a^5}-\frac{x^4}{100 a}+\frac{1}{25} x^5 \log (1-a x)-\frac{x^5}{125} \]
Antiderivative was successfully verified.
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Rule 6591
Rule 2395
Rule 43
Rubi steps
\begin{align*} \int x^4 \text{Li}_2(a x) \, dx &=\frac{1}{5} x^5 \text{Li}_2(a x)+\frac{1}{5} \int x^4 \log (1-a x) \, dx\\ &=\frac{1}{25} x^5 \log (1-a x)+\frac{1}{5} x^5 \text{Li}_2(a x)+\frac{1}{25} a \int \frac{x^5}{1-a x} \, dx\\ &=\frac{1}{25} x^5 \log (1-a x)+\frac{1}{5} x^5 \text{Li}_2(a x)+\frac{1}{25} a \int \left (-\frac{1}{a^5}-\frac{x}{a^4}-\frac{x^2}{a^3}-\frac{x^3}{a^2}-\frac{x^4}{a}-\frac{1}{a^5 (-1+a x)}\right ) \, dx\\ &=-\frac{x}{25 a^4}-\frac{x^2}{50 a^3}-\frac{x^3}{75 a^2}-\frac{x^4}{100 a}-\frac{x^5}{125}-\frac{\log (1-a x)}{25 a^5}+\frac{1}{25} x^5 \log (1-a x)+\frac{1}{5} x^5 \text{Li}_2(a x)\\ \end{align*}
Mathematica [A] time = 0.04193, size = 73, normalized size = 0.85 \[ \frac{300 a^5 x^5 \text{PolyLog}(2,a x)-a x \left (12 a^4 x^4+15 a^3 x^3+20 a^2 x^2+30 a x+60\right )+60 \left (a^5 x^5-1\right ) \log (1-a x)}{1500 a^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 76, normalized size = 0.9 \begin{align*} -{\frac{x}{25\,{a}^{4}}}-{\frac{\ln \left ( -ax+1 \right ) }{25\,{a}^{5}}}-{\frac{{x}^{2}}{50\,{a}^{3}}}+{\frac{{x}^{5}{\it polylog} \left ( 2,ax \right ) }{5}}+{\frac{{x}^{5}\ln \left ( -ax+1 \right ) }{25}}+{\frac{137}{1500\,{a}^{5}}}-{\frac{{x}^{4}}{100\,a}}-{\frac{{x}^{3}}{75\,{a}^{2}}}-{\frac{{x}^{5}}{125}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.964112, size = 97, normalized size = 1.13 \begin{align*} \frac{300 \, a^{5} x^{5}{\rm Li}_2\left (a x\right ) - 12 \, a^{5} x^{5} - 15 \, a^{4} x^{4} - 20 \, a^{3} x^{3} - 30 \, a^{2} x^{2} - 60 \, a x + 60 \,{\left (a^{5} x^{5} - 1\right )} \log \left (-a x + 1\right )}{1500 \, a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.68893, size = 177, normalized size = 2.06 \begin{align*} \frac{300 \, a^{5} x^{5}{\rm Li}_2\left (a x\right ) - 12 \, a^{5} x^{5} - 15 \, a^{4} x^{4} - 20 \, a^{3} x^{3} - 30 \, a^{2} x^{2} - 60 \, a x + 60 \,{\left (a^{5} x^{5} - 1\right )} \log \left (-a x + 1\right )}{1500 \, a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 15.264, size = 66, normalized size = 0.77 \begin{align*} \begin{cases} - \frac{x^{5} \operatorname{Li}_{1}\left (a x\right )}{25} + \frac{x^{5} \operatorname{Li}_{2}\left (a x\right )}{5} - \frac{x^{5}}{125} - \frac{x^{4}}{100 a} - \frac{x^{3}}{75 a^{2}} - \frac{x^{2}}{50 a^{3}} - \frac{x}{25 a^{4}} + \frac{\operatorname{Li}_{1}\left (a x\right )}{25 a^{5}} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \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^{4}{\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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