Optimal. Leaf size=253 \[ -\frac{1}{9} x^3 \text{PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )+\frac{1}{3} x^3 \text{PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )+\frac{b n \text{PolyLog}(2,e x)}{27 e^3}+\frac{2}{27} b n x^3 \text{PolyLog}(2,e x)-\frac{1}{9} b n x^3 \text{PolyLog}(3,e x)+\frac{x \left (a+b \log \left (c x^n\right )\right )}{27 e^2}+\frac{\log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{27 e^3}-\frac{1}{27} x^3 \log (1-e x) \left (a+b \log \left (c x^n\right )\right )+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{54 e}+\frac{1}{81} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{2 b n x}{27 e^2}-\frac{b n \log (1-e x)}{27 e^3}-\frac{b n x^2}{36 e}+\frac{1}{27} b n x^3 \log (1-e x)-\frac{4}{243} b n x^3 \]
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Rubi [A] time = 0.254357, antiderivative size = 253, normalized size of antiderivative = 1., 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 = {2385, 2395, 43, 2376, 2391, 6591} \[ -\frac{1}{9} x^3 \text{PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )+\frac{1}{3} x^3 \text{PolyLog}(3,e x) \left (a+b \log \left (c x^n\right )\right )+\frac{b n \text{PolyLog}(2,e x)}{27 e^3}+\frac{2}{27} b n x^3 \text{PolyLog}(2,e x)-\frac{1}{9} b n x^3 \text{PolyLog}(3,e x)+\frac{x \left (a+b \log \left (c x^n\right )\right )}{27 e^2}+\frac{\log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{27 e^3}-\frac{1}{27} x^3 \log (1-e x) \left (a+b \log \left (c x^n\right )\right )+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{54 e}+\frac{1}{81} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{2 b n x}{27 e^2}-\frac{b n \log (1-e x)}{27 e^3}-\frac{b n x^2}{36 e}+\frac{1}{27} b n x^3 \log (1-e x)-\frac{4}{243} b n x^3 \]
Antiderivative was successfully verified.
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Rule 2385
Rule 2395
Rule 43
Rule 2376
Rule 2391
Rule 6591
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*}
Mathematica [F] time = 0.135257, size = 0, normalized size = 0. \[ \int x^2 \left (a+b \log \left (c x^n\right )\right ) \text{PolyLog}(3,e x) \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.349, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ){\it polylog} \left ( 3,ex \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{162} \, b{\left (\frac{6 \,{\left (3 \, e^{3} x^{3} \log \left (x^{n}\right ) -{\left (2 \, e^{3} n - 3 \, e^{3} \log \left (c\right )\right )} x^{3}\right )}{\rm Li}_2\left (e x\right ) - 6 \,{\left ({\left (e^{3} n - e^{3} \log \left (c\right )\right )} x^{3} - n \log \left (x\right )\right )} \log \left (-e x + 1\right ) -{\left (2 \, e^{3} x^{3} + 3 \, e^{2} x^{2} + 6 \, e x - 6 \,{\left (e^{3} x^{3} - 1\right )} \log \left (-e x + 1\right )\right )} \log \left (x^{n}\right ) - 18 \,{\left (3 \, e^{3} x^{3} \log \left (x^{n}\right ) -{\left (e^{3} n - 3 \, e^{3} \log \left (c\right )\right )} x^{3}\right )}{\rm Li}_{3}(e x)}{e^{3}} - 162 \, \int -\frac{e^{2} n x^{2} + 2 \,{\left (4 \, e^{3} n - 3 \, e^{3} \log \left (c\right )\right )} x^{3} + 3 \, e n x - 6 \, n \log \left (x\right ) - 6 \, n}{162 \,{\left (e^{3} x - e^{2}\right )}}\,{d x}\right )} - \frac{{\left (18 \, e^{3} x^{3}{\rm Li}_2\left (e x\right ) - 54 \, e^{3} x^{3}{\rm Li}_{3}(e x) - 2 \, e^{3} x^{3} - 3 \, e^{2} x^{2} - 6 \, e x + 6 \,{\left (e^{3} x^{3} - 1\right )} \log \left (-e x + 1\right )\right )} a}{162 \, e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 0.975218, size = 738, normalized size = 2.92 \begin{align*} -\frac{4 \,{\left (4 \, b e^{3} n - 3 \, a e^{3}\right )} x^{3} + 9 \,{\left (3 \, b e^{2} n - 2 \, a e^{2}\right )} x^{2} + 36 \,{\left (2 \, b e n - a e\right )} x + 36 \,{\left (3 \, b e^{3} n x^{3} \log \left (x\right ) + 3 \, b e^{3} x^{3} \log \left (c\right ) -{\left (2 \, b e^{3} n - 3 \, a e^{3}\right )} x^{3} - b n\right )}{\rm \%iint}\left (e, x, -\frac{\log \left (-e x + 1\right )}{e}, -\frac{\log \left (-e x + 1\right )}{x}\right ) - 36 \,{\left ({\left (b e^{3} n - a e^{3}\right )} x^{3} - b n + a\right )} \log \left (-e x + 1\right ) - 6 \,{\left (2 \, b e^{3} x^{3} + 3 \, b e^{2} x^{2} + 6 \, b e x - 6 \,{\left (b e^{3} x^{3} - b\right )} \log \left (-e x + 1\right )\right )} \log \left (c\right ) - 6 \,{\left (2 \, b e^{3} n x^{3} + 3 \, b e^{2} n x^{2} + 6 \, b e n x - 6 \,{\left (b e^{3} n x^{3} - b n\right )} \log \left (-e x + 1\right )\right )} \log \left (x\right ) - 108 \,{\left (3 \, b e^{3} n x^{3} \log \left (x\right ) + 3 \, b e^{3} x^{3} \log \left (c\right ) -{\left (b e^{3} n - 3 \, a e^{3}\right )} x^{3}\right )}{\rm polylog}\left (3, e x\right )}{972 \, e^{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} \left (a + b \log{\left (c x^{n} \right )}\right ) \operatorname{Li}_{3}\left (e 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{\left (b \log \left (c x^{n}\right ) + a\right )} x^{2}{\rm Li}_{3}(e x)\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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