Optimal. Leaf size=253 \[ -\frac {2 b n x}{27 e^2}-\frac {b n x^2}{36 e}-\frac {4}{243} b n x^3+\frac {x \left (a+b \log \left (c x^n\right )\right )}{27 e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{54 e}+\frac {1}{81} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {b n \log (1-e x)}{27 e^3}+\frac {1}{27} b n x^3 \log (1-e x)+\frac {\left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{27 e^3}-\frac {1}{27} x^3 \left (a+b \log \left (c x^n\right )\right ) \log (1-e x)+\frac {b n \text {Li}_2(e x)}{27 e^3}+\frac {2}{27} b n x^3 \text {Li}_2(e x)-\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(e x)-\frac {1}{9} b n x^3 \text {Li}_3(e x)+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(e x) \]
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Rubi [A]
time = 0.18, antiderivative size = 253, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {2432, 2442,
45, 2423, 2438, 6726} \begin {gather*} -\frac {1}{9} x^3 \text {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )+\frac {1}{3} x^3 \text {PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )+\frac {b n \text {PolyLog}(2,e x)}{27 e^3}+\frac {2}{27} b n x^3 \text {PolyLog}(2,e x)-\frac {1}{9} b n x^3 \text {PolyLog}(3,e x)+\frac {\log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{27 e^3}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{27 e^2}-\frac {1}{27} 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 )}{54 e}+\frac {1}{81} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {b n \log (1-e x)}{27 e^3}-\frac {2 b n x}{27 e^2}+\frac {1}{27} b n x^3 \log (1-e x)-\frac {b n x^2}{36 e}-\frac {4}{243} b n x^3 \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2423
Rule 2432
Rule 2438
Rule 2442
Rule 6726
Rubi steps
\begin {align*} \int x^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(e x) \, dx &=-\frac {1}{9} b n x^3 \text {Li}_3(e x)+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(e x)-\frac {1}{3} \int x^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(e x) \, dx+\frac {1}{9} (b n) \int x^2 \text {Li}_2(e x) \, dx\\ &=\frac {2}{27} b n x^3 \text {Li}_2(e x)-\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(e x)-\frac {1}{9} b n x^3 \text {Li}_3(e x)+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(e x)-\frac {1}{9} \int x^2 \left (a+b \log \left (c x^n\right )\right ) \log (1-e x) \, dx+2 \left (\frac {1}{27} (b n) \int x^2 \log (1-e x) \, dx\right )\\ &=\frac {x \left (a+b \log \left (c x^n\right )\right )}{27 e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{54 e}+\frac {1}{81} x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {\left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{27 e^3}-\frac {1}{27} x^3 \left (a+b \log \left (c x^n\right )\right ) \log (1-e x)+\frac {2}{27} b n x^3 \text {Li}_2(e x)-\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(e x)-\frac {1}{9} b n x^3 \text {Li}_3(e x)+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(e x)+\frac {1}{9} (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+2 \left (\frac {1}{81} b n x^3 \log (1-e x)+\frac {1}{81} (b e n) \int \frac {x^3}{1-e x} \, dx\right )\\ &=-\frac {b n x}{27 e^2}-\frac {b n x^2}{108 e}-\frac {1}{243} b n x^3+\frac {x \left (a+b \log \left (c x^n\right )\right )}{27 e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{54 e}+\frac {1}{81} x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {\left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{27 e^3}-\frac {1}{27} x^3 \left (a+b \log \left (c x^n\right )\right ) \log (1-e x)+\frac {2}{27} b n x^3 \text {Li}_2(e x)-\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(e x)-\frac {1}{9} b n x^3 \text {Li}_3(e x)+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(e x)+\frac {1}{27} (b n) \int x^2 \log (1-e x) \, dx-\frac {(b n) \int \frac {\log (1-e x)}{x} \, dx}{27 e^3}+2 \left (\frac {1}{81} b n x^3 \log (1-e x)+\frac {1}{81} (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\right )\\ &=-\frac {b n x}{27 e^2}-\frac {b n x^2}{108 e}-\frac {1}{243} b n x^3+\frac {x \left (a+b \log \left (c x^n\right )\right )}{27 e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{54 e}+\frac {1}{81} x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{81} b n x^3 \log (1-e x)+\frac {\left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{27 e^3}-\frac {1}{27} x^3 \left (a+b \log \left (c x^n\right )\right ) \log (1-e x)+2 \left (-\frac {b n x}{81 e^2}-\frac {b n x^2}{162 e}-\frac {1}{243} b n x^3-\frac {b n \log (1-e x)}{81 e^3}+\frac {1}{81} b n x^3 \log (1-e x)\right )+\frac {b n \text {Li}_2(e x)}{27 e^3}+\frac {2}{27} b n x^3 \text {Li}_2(e x)-\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(e x)-\frac {1}{9} b n x^3 \text {Li}_3(e x)+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(e x)+\frac {1}{81} (b e n) \int \frac {x^3}{1-e x} \, dx\\ &=-\frac {b n x}{27 e^2}-\frac {b n x^2}{108 e}-\frac {1}{243} b n x^3+\frac {x \left (a+b \log \left (c x^n\right )\right )}{27 e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{54 e}+\frac {1}{81} x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{81} b n x^3 \log (1-e x)+\frac {\left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{27 e^3}-\frac {1}{27} x^3 \left (a+b \log \left (c x^n\right )\right ) \log (1-e x)+2 \left (-\frac {b n x}{81 e^2}-\frac {b n x^2}{162 e}-\frac {1}{243} b n x^3-\frac {b n \log (1-e x)}{81 e^3}+\frac {1}{81} b n x^3 \log (1-e x)\right )+\frac {b n \text {Li}_2(e x)}{27 e^3}+\frac {2}{27} b n x^3 \text {Li}_2(e x)-\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(e x)-\frac {1}{9} b n x^3 \text {Li}_3(e x)+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(e x)+\frac {1}{81} (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}{81 e^2}-\frac {5 b n x^2}{324 e}-\frac {2}{243} b n x^3+\frac {x \left (a+b \log \left (c x^n\right )\right )}{27 e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{54 e}+\frac {1}{81} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {b n \log (1-e x)}{81 e^3}+\frac {1}{81} b n x^3 \log (1-e x)+\frac {\left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{27 e^3}-\frac {1}{27} x^3 \left (a+b \log \left (c x^n\right )\right ) \log (1-e x)+2 \left (-\frac {b n x}{81 e^2}-\frac {b n x^2}{162 e}-\frac {1}{243} b n x^3-\frac {b n \log (1-e x)}{81 e^3}+\frac {1}{81} b n x^3 \log (1-e x)\right )+\frac {b n \text {Li}_2(e x)}{27 e^3}+\frac {2}{27} b n x^3 \text {Li}_2(e x)-\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(e x)-\frac {1}{9} b n x^3 \text {Li}_3(e x)+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(e x)\\ \end {align*}
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Mathematica [F]
time = 0.10, size = 0, normalized size = 0.00 \begin {gather*} \int x^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(e x) \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int x^{2} \left (a +b \ln \left (c \,x^{n}\right )\right ) \polylog \left (3, e x \right )\, dx\]
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 = 0.38, size = 274, normalized size = 1.08 \begin {gather*} -\frac {1}{972} \, {\left (4 \, {\left (4 \, b n - 3 \, a\right )} x^{3} e^{3} + 9 \, {\left (3 \, b n - 2 \, a\right )} x^{2} e^{2} + 36 \, {\left (2 \, b n - a\right )} x e - 36 \, {\left ({\left (2 \, b n - 3 \, a\right )} x^{3} e^{3} + b n\right )} {\rm Li}_2\left (x e\right ) - 36 \, {\left ({\left (b n - a\right )} x^{3} e^{3} - b n + a\right )} \log \left (-x e + 1\right ) + 6 \, {\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 ) + 6 \, {\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 ) - 108 \, {\left (3 \, b n x^{3} e^{3} \log \left (x\right ) + 3 \, b x^{3} e^{3} \log \left (c\right ) - {\left (b n - 3 \, a\right )} x^{3} e^{3}\right )} {\rm polylog}\left (3, x e\right )\right )} e^{\left (-3\right )} \end {gather*}
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 \begin {gather*} \int x^{2} \left (a + b \log {\left (c x^{n} \right )}\right ) \operatorname {Li}_{3}\left (e x\right )\, dx \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(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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