Optimal. Leaf size=86 \[ -\frac {\log (1-a x)}{25 a^5}-\frac {x}{25 a^4}-\frac {x^2}{50 a^3}-\frac {x^3}{75 a^2}+\frac {1}{5} x^5 \text {Li}_2(a x)+\frac {1}{25} x^5 \log (1-a x)-\frac {x^4}{100 a}-\frac {x^5}{125} \]
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Rubi [A] time = 0.05, antiderivative size = 86, normalized size of antiderivative = 1.00, 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 43
Rule 2395
Rule 6591
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*}
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Mathematica [A] time = 0.04, size = 73, normalized size = 0.85 \[ \frac {300 a^5 x^5 \text {Li}_2(a x)+60 \left (a^5 x^5-1\right ) \log (1-a x)-a x \left (12 a^4 x^4+15 a^3 x^3+20 a^2 x^2+30 a x+60\right )}{1500 a^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.21, size = 72, normalized size = 0.84 \[ \frac {300 \, a^{5} x^{5} {\rm Li}_2\left (a x\right ) - 12 \, a^{5} x^{5} - 15 \, a^{4} x^{4} - 20 \, a^{3} x^{3} - 30 \, a^{2} x^{2} - 60 \, a x + 60 \, {\left (a^{5} x^{5} - 1\right )} \log \left (-a x + 1\right )}{1500 \, a^{5}} \]
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 \[ \int x^{4} {\rm Li}_2\left (a x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 76, normalized size = 0.88 \[ -\frac {x}{25 a^{4}}-\frac {x^{5}}{125}-\frac {x^{4}}{100 a}-\frac {x^{3}}{75 a^{2}}-\frac {x^{2}}{50 a^{3}}-\frac {\ln \left (-a x +1\right )}{25 a^{5}}+\frac {x^{5} \ln \left (-a x +1\right )}{25}+\frac {x^{5} \polylog \left (2, a x \right )}{5}+\frac {137}{1500 a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 72, normalized size = 0.84 \[ \frac {300 \, a^{5} x^{5} {\rm Li}_2\left (a x\right ) - 12 \, a^{5} x^{5} - 15 \, a^{4} x^{4} - 20 \, a^{3} x^{3} - 30 \, a^{2} x^{2} - 60 \, a x + 60 \, {\left (a^{5} x^{5} - 1\right )} \log \left (-a x + 1\right )}{1500 \, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.31, size = 69, normalized size = 0.80 \[ \frac {x^5\,\ln \left (1-a\,x\right )}{25}-\frac {\ln \left (a\,x-1\right )}{25\,a^5}-\frac {x}{25\,a^4}-\frac {x^5}{125}+\frac {x^5\,\mathrm {polylog}\left (2,a\,x\right )}{5}-\frac {x^4}{100\,a}-\frac {x^3}{75\,a^2}-\frac {x^2}{50\,a^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 9.68, size = 66, normalized size = 0.77 \[ \begin {cases} - \frac {x^{5} \operatorname {Li}_{1}\left (a x\right )}{25} + \frac {x^{5} \operatorname {Li}_{2}\left (a x\right )}{5} - \frac {x^{5}}{125} - \frac {x^{4}}{100 a} - \frac {x^{3}}{75 a^{2}} - \frac {x^{2}}{50 a^{3}} - \frac {x}{25 a^{4}} + \frac {\operatorname {Li}_{1}\left (a x\right )}{25 a^{5}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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