Optimal. Leaf size=217 \[ \frac {5 b n x}{27 e^2}+\frac {7 b n x^2}{108 e}+\frac {1}{27} b n x^3-\frac {x \left (a+b \log \left (c x^n\right )\right )}{9 e^2}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{18 e}-\frac {1}{27} x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {2 b n \log (1-e x)}{27 e^3}-\frac {2}{27} b n x^3 \log (1-e x)-\frac {\left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{9 e^3}+\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right ) \log (1-e x)-\frac {b n \text {Li}_2(e x)}{9 e^3}-\frac {1}{9} b n x^3 \text {Li}_2(e x)+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(e x) \]
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Rubi [A]
time = 0.13, antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2432, 2442,
45, 2423, 2438} \begin {gather*} \frac {1}{3} x^3 \text {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )-\frac {b n \text {PolyLog}(2,e x)}{9 e^3}-\frac {1}{9} b n x^3 \text {PolyLog}(2,e x)-\frac {\log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{9 e^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{9 e^2}+\frac {1}{9} x^3 \log (1-e x) \left (a+b \log \left (c x^n\right )\right )-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{18 e}-\frac {1}{27} x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {2 b n \log (1-e x)}{27 e^3}+\frac {5 b n x}{27 e^2}-\frac {2}{27} b n x^3 \log (1-e x)+\frac {7 b n x^2}{108 e}+\frac {1}{27} b n x^3 \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2423
Rule 2432
Rule 2438
Rule 2442
Rubi steps
\begin {align*} \int x^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(e x) \, dx &=-\frac {1}{9} b n x^3 \text {Li}_2(e x)+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(e x)+\frac {1}{3} \int x^2 \left (a+b \log \left (c x^n\right )\right ) \log (1-e x) \, dx-\frac {1}{9} (b n) \int x^2 \log (1-e x) \, dx\\ &=-\frac {x \left (a+b \log \left (c x^n\right )\right )}{9 e^2}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{18 e}-\frac {1}{27} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{27} b n x^3 \log (1-e x)-\frac {\left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{9 e^3}+\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right ) \log (1-e x)-\frac {1}{9} b n x^3 \text {Li}_2(e x)+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(e x)-\frac {1}{3} (b n) \int \left (-\frac {1}{3 e^2}-\frac {x}{6 e}-\frac {x^2}{9}-\frac {\log (1-e x)}{3 e^3 x}+\frac {1}{3} x^2 \log (1-e x)\right ) \, dx-\frac {1}{27} (b e n) \int \frac {x^3}{1-e x} \, dx\\ &=\frac {b n x}{9 e^2}+\frac {b n x^2}{36 e}+\frac {1}{81} b n x^3-\frac {x \left (a+b \log \left (c x^n\right )\right )}{9 e^2}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{18 e}-\frac {1}{27} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{27} b n x^3 \log (1-e x)-\frac {\left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{9 e^3}+\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right ) \log (1-e x)-\frac {1}{9} b n x^3 \text {Li}_2(e x)+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(e x)-\frac {1}{9} (b n) \int x^2 \log (1-e x) \, dx+\frac {(b n) \int \frac {\log (1-e x)}{x} \, dx}{9 e^3}-\frac {1}{27} (b e n) \int \left (-\frac {1}{e^3}-\frac {x}{e^2}-\frac {x^2}{e}-\frac {1}{e^3 (-1+e x)}\right ) \, dx\\ &=\frac {4 b n x}{27 e^2}+\frac {5 b n x^2}{108 e}+\frac {2}{81} b n x^3-\frac {x \left (a+b \log \left (c x^n\right )\right )}{9 e^2}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{18 e}-\frac {1}{27} x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {b n \log (1-e x)}{27 e^3}-\frac {2}{27} b n x^3 \log (1-e x)-\frac {\left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{9 e^3}+\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right ) \log (1-e x)-\frac {b n \text {Li}_2(e x)}{9 e^3}-\frac {1}{9} b n x^3 \text {Li}_2(e x)+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(e x)-\frac {1}{27} (b e n) \int \frac {x^3}{1-e x} \, dx\\ &=\frac {4 b n x}{27 e^2}+\frac {5 b n x^2}{108 e}+\frac {2}{81} b n x^3-\frac {x \left (a+b \log \left (c x^n\right )\right )}{9 e^2}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{18 e}-\frac {1}{27} x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {b n \log (1-e x)}{27 e^3}-\frac {2}{27} b n x^3 \log (1-e x)-\frac {\left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{9 e^3}+\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right ) \log (1-e x)-\frac {b n \text {Li}_2(e x)}{9 e^3}-\frac {1}{9} b n x^3 \text {Li}_2(e x)+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(e x)-\frac {1}{27} (b e n) \int \left (-\frac {1}{e^3}-\frac {x}{e^2}-\frac {x^2}{e}-\frac {1}{e^3 (-1+e x)}\right ) \, dx\\ &=\frac {5 b n x}{27 e^2}+\frac {7 b n x^2}{108 e}+\frac {1}{27} b n x^3-\frac {x \left (a+b \log \left (c x^n\right )\right )}{9 e^2}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{18 e}-\frac {1}{27} x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {2 b n \log (1-e x)}{27 e^3}-\frac {2}{27} b n x^3 \log (1-e x)-\frac {\left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{9 e^3}+\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right ) \log (1-e x)-\frac {b n \text {Li}_2(e x)}{9 e^3}-\frac {1}{9} b n x^3 \text {Li}_2(e x)+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(e x)\\ \end {align*}
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Mathematica [A]
time = 0.39, size = 196, normalized size = 0.90 \begin {gather*} \frac {\left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \left (-e x \left (6+3 e x+2 e^2 x^2\right )+6 \left (-1+e^3 x^3\right ) \log (1-e x)+18 e^3 x^3 \text {Li}_2(e x)\right )}{54 e^3}+\frac {b n \left (20 e x+7 e^2 x^2+4 e^3 x^3+8 \log (1-e x)-8 e^3 x^3 \log (1-e x)+2 \log (x) \left (-e x \left (6+3 e x+2 e^2 x^2\right )+6 \left (-1+e^3 x^3\right ) \log (1-e x)\right )+12 \left (-1-e^3 x^3+3 e^3 x^3 \log (x)\right ) \text {Li}_2(e x)\right )}{108 e^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int x^{2} \left (a +b \ln \left (c \,x^{n}\right )\right ) \polylog \left (2, e x \right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 230, normalized size = 1.06 \begin {gather*} \frac {1}{108} \, {\left (4 \, {\left (b n - a\right )} x^{3} e^{3} + {\left (7 \, b n - 6 \, a\right )} x^{2} e^{2} + 4 \, {\left (5 \, b n - 3 \, a\right )} x e - 12 \, {\left ({\left (b n - 3 \, a\right )} x^{3} e^{3} + b n\right )} {\rm Li}_2\left (x e\right ) - 4 \, {\left ({\left (2 \, b n - 3 \, a\right )} x^{3} e^{3} - 2 \, b n + 3 \, a\right )} \log \left (-x e + 1\right ) + 2 \, {\left (18 \, b x^{3} {\rm Li}_2\left (x e\right ) e^{3} - 2 \, b x^{3} e^{3} - 3 \, b x^{2} e^{2} - 6 \, b x e + 6 \, {\left (b x^{3} e^{3} - b\right )} \log \left (-x e + 1\right )\right )} \log \left (c\right ) + 2 \, {\left (18 \, b n x^{3} {\rm Li}_2\left (x e\right ) e^{3} - 2 \, b n x^{3} e^{3} - 3 \, b n x^{2} e^{2} - 6 \, b n x e + 6 \, {\left (b n x^{3} e^{3} - b n\right )} \log \left (-x e + 1\right )\right )} \log \left (x\right )\right )} e^{\left (-3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 76.59, size = 250, normalized size = 1.15 \begin {gather*} \begin {cases} - \frac {a x^{3} \operatorname {Li}_{1}\left (e x\right )}{9} + \frac {a x^{3} \operatorname {Li}_{2}\left (e x\right )}{3} - \frac {a x^{3}}{27} - \frac {a x^{2}}{18 e} - \frac {a x}{9 e^{2}} + \frac {a \operatorname {Li}_{1}\left (e x\right )}{9 e^{3}} + \frac {2 b n x^{3} \operatorname {Li}_{1}\left (e x\right )}{27} - \frac {b n x^{3} \operatorname {Li}_{2}\left (e x\right )}{9} + \frac {b n x^{3}}{27} - \frac {b x^{3} \log {\left (c x^{n} \right )} \operatorname {Li}_{1}\left (e x\right )}{9} + \frac {b x^{3} \log {\left (c x^{n} \right )} \operatorname {Li}_{2}\left (e x\right )}{3} - \frac {b x^{3} \log {\left (c x^{n} \right )}}{27} + \frac {7 b n x^{2}}{108 e} - \frac {b x^{2} \log {\left (c x^{n} \right )}}{18 e} + \frac {5 b n x}{27 e^{2}} - \frac {b x \log {\left (c x^{n} \right )}}{9 e^{2}} - \frac {2 b n \operatorname {Li}_{1}\left (e x\right )}{27 e^{3}} - \frac {b n \operatorname {Li}_{2}\left (e x\right )}{9 e^{3}} + \frac {b \log {\left (c x^{n} \right )} \operatorname {Li}_{1}\left (e x\right )}{9 e^{3}} & \text {for}\: e \neq 0 \\0 & \text {otherwise} \end {cases} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,\mathrm {polylog}\left (2,e\,x\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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