Optimal. Leaf size=78 \[ \frac {\log (1-a x)}{27 a^3}+\frac {x}{27 a^2}-\frac {1}{9} x^3 \text {Li}_2(a x)+\frac {1}{3} x^3 \text {Li}_3(a x)-\frac {1}{27} x^3 \log (1-a x)+\frac {x^2}{54 a}+\frac {x^3}{81} \]
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Rubi [A] time = 0.05, antiderivative size = 78, normalized size of antiderivative = 1.00, 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 43
Rule 2395
Rule 6591
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*}
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Mathematica [A] time = 0.01, size = 78, normalized size = 1.00 \[ \frac {-18 a^3 x^3 \text {Li}_2(a x)+54 a^3 x^3 \text {Li}_3(a x)+2 a^3 x^3-6 a^3 x^3 \log (1-a x)+3 a^2 x^2+6 a x+6 \log (1-a x)}{162 a^3} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.95, size = 69, normalized size = 0.88 \[ -\frac {18 \, a^{3} x^{3} {\rm Li}_2\left (a x\right ) - 54 \, a^{3} x^{3} {\rm polylog}\left (3, a x\right ) - 2 \, a^{3} x^{3} - 3 \, a^{2} x^{2} - 6 \, a x + 6 \, {\left (a^{3} x^{3} - 1\right )} \log \left (-a x + 1\right )}{162 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} {\rm Li}_{3}(a x)\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 69, normalized size = 0.88 \[ \frac {\frac {a x \left (4 a^{2} x^{2}+6 a x +12\right )}{324}+\frac {\left (-4 a^{3} x^{3}+4\right ) \ln \left (-a x +1\right )}{108}-\frac {a^{3} x^{3} \polylog \left (2, a x \right )}{9}+\frac {a^{3} x^{3} \polylog \left (3, a x \right )}{3}}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 69, normalized size = 0.88 \[ -\frac {18 \, a^{3} x^{3} {\rm Li}_2\left (a x\right ) - 54 \, a^{3} x^{3} {\rm Li}_{3}(a x) - 2 \, a^{3} x^{3} - 3 \, a^{2} x^{2} - 6 \, a x + 6 \, {\left (a^{3} x^{3} - 1\right )} \log \left (-a x + 1\right )}{162 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.95, size = 63, normalized size = 0.81 \[ \frac {\ln \left (a\,x-1\right )}{27\,a^3}-\frac {x^3\,\ln \left (1-a\,x\right )}{27}+\frac {x}{27\,a^2}+\frac {x^3}{81}-\frac {x^3\,\mathrm {polylog}\left (2,a\,x\right )}{9}+\frac {x^3\,\mathrm {polylog}\left (3,a\,x\right )}{3}+\frac {x^2}{54\,a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \operatorname {Li}_{3}\left (a x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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