Optimal. Leaf size=78 \[ -\frac{1}{9} x^3 \text{PolyLog}(2,a x)+\frac{1}{3} x^3 \text{PolyLog}(3,a x)+\frac{x}{27 a^2}+\frac{\log (1-a x)}{27 a^3}+\frac{x^2}{54 a}-\frac{1}{27} x^3 \log (1-a x)+\frac{x^3}{81} \]
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Rubi [A] time = 0.0506273, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6591, 2395, 43} \[ -\frac{1}{9} x^3 \text{PolyLog}(2,a x)+\frac{1}{3} x^3 \text{PolyLog}(3,a x)+\frac{x}{27 a^2}+\frac{\log (1-a x)}{27 a^3}+\frac{x^2}{54 a}-\frac{1}{27} x^3 \log (1-a x)+\frac{x^3}{81} \]
Antiderivative was successfully verified.
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Rule 6591
Rule 2395
Rule 43
Rubi steps
\begin{align*} \int x^2 \text{Li}_3(a x) \, dx &=\frac{1}{3} x^3 \text{Li}_3(a x)-\frac{1}{3} \int x^2 \text{Li}_2(a x) \, dx\\ &=-\frac{1}{9} x^3 \text{Li}_2(a x)+\frac{1}{3} x^3 \text{Li}_3(a x)-\frac{1}{9} \int x^2 \log (1-a x) \, dx\\ &=-\frac{1}{27} x^3 \log (1-a x)-\frac{1}{9} x^3 \text{Li}_2(a x)+\frac{1}{3} x^3 \text{Li}_3(a x)-\frac{1}{27} a \int \frac{x^3}{1-a x} \, dx\\ &=-\frac{1}{27} x^3 \log (1-a x)-\frac{1}{9} x^3 \text{Li}_2(a x)+\frac{1}{3} x^3 \text{Li}_3(a x)-\frac{1}{27} a \int \left (-\frac{1}{a^3}-\frac{x}{a^2}-\frac{x^2}{a}-\frac{1}{a^3 (-1+a x)}\right ) \, dx\\ &=\frac{x}{27 a^2}+\frac{x^2}{54 a}+\frac{x^3}{81}+\frac{\log (1-a x)}{27 a^3}-\frac{1}{27} x^3 \log (1-a x)-\frac{1}{9} x^3 \text{Li}_2(a x)+\frac{1}{3} x^3 \text{Li}_3(a x)\\ \end{align*}
Mathematica [A] time = 0.0113884, size = 78, normalized size = 1. \[ \frac{-18 a^3 x^3 \text{PolyLog}(2,a x)+54 a^3 x^3 \text{PolyLog}(3,a x)+2 a^3 x^3+3 a^2 x^2-6 a^3 x^3 \log (1-a x)+6 a x+6 \log (1-a x)}{162 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.164, size = 69, normalized size = 0.9 \begin{align*}{\frac{1}{{a}^{3}} \left ({\frac{xa \left ( 4\,{a}^{2}{x}^{2}+6\,ax+12 \right ) }{324}}+{\frac{ \left ( -4\,{x}^{3}{a}^{3}+4 \right ) \ln \left ( -ax+1 \right ) }{108}}-{\frac{{x}^{3}{a}^{3}{\it polylog} \left ( 2,ax \right ) }{9}}+{\frac{{x}^{3}{a}^{3}{\it polylog} \left ( 3,ax \right ) }{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.995622, size = 93, normalized size = 1.19 \begin{align*} -\frac{18 \, a^{3} x^{3}{\rm Li}_2\left (a x\right ) - 54 \, a^{3} x^{3}{\rm Li}_{3}(a x) - 2 \, a^{3} x^{3} - 3 \, a^{2} x^{2} - 6 \, a x + 6 \,{\left (a^{3} x^{3} - 1\right )} \log \left (-a x + 1\right )}{162 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.67641, size = 224, normalized size = 2.87 \begin{align*} -\frac{18 \, a^{3} x^{3}{\rm \%iint}\left (a, x, -\frac{\log \left (-a x + 1\right )}{a}, -\frac{\log \left (-a x + 1\right )}{x}\right ) - 54 \, a^{3} x^{3}{\rm polylog}\left (3, a x\right ) - 2 \, a^{3} x^{3} - 3 \, a^{2} x^{2} - 6 \, a x + 6 \,{\left (a^{3} x^{3} - 1\right )} \log \left (-a x + 1\right )}{162 \, a^{3}} \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^{2} \operatorname{Li}_{3}\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 x^{2}{\rm Li}_{3}(a x)\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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