Optimal. Leaf size=253 \[ \frac {\log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{27 e^3}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{27 e^2}-\frac {1}{9} x^3 \text {Li}_2(e x) \left (a+b \log \left (c x^n\right )\right )+\frac {1}{3} x^3 \text {Li}_3(e x) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{27} x^3 \log (1-e x) \left (a+b \log \left (c x^n\right )\right )+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{54 e}+\frac {1}{81} x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {b n \text {Li}_2(e x)}{27 e^3}-\frac {b n \log (1-e x)}{27 e^3}-\frac {2 b n x}{27 e^2}+\frac {2}{27} b n x^3 \text {Li}_2(e x)-\frac {1}{9} b n x^3 \text {Li}_3(e x)+\frac {1}{27} b n x^3 \log (1-e x)-\frac {b n x^2}{36 e}-\frac {4}{243} b n x^3 \]
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Rubi [A] time = 0.25, 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 = {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 43
Rule 2376
Rule 2385
Rule 2391
Rule 2395
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*}
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Mathematica [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int x^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(e x) \, dx \]
Verification is Not applicable to the result.
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fricas [C] time = 0.79, size = 296, normalized size = 1.17 \[ -\frac {4 \, {\left (4 \, b e^{3} n - 3 \, a e^{3}\right )} x^{3} + 9 \, {\left (3 \, b e^{2} n - 2 \, a e^{2}\right )} x^{2} + 36 \, {\left (2 \, b e n - a e\right )} x - 36 \, {\left ({\left (2 \, b e^{3} n - 3 \, a e^{3}\right )} x^{3} + b n\right )} {\rm Li}_2\left (e x\right ) - 36 \, {\left ({\left (b e^{3} n - a e^{3}\right )} x^{3} - b n + a\right )} \log \left (-e x + 1\right ) + 6 \, {\left (18 \, b e^{3} x^{3} {\rm Li}_2\left (e x\right ) - 2 \, b e^{3} x^{3} - 3 \, b e^{2} x^{2} - 6 \, b e x + 6 \, {\left (b e^{3} x^{3} - b\right )} \log \left (-e x + 1\right )\right )} \log \relax (c) + 6 \, {\left (18 \, b e^{3} n x^{3} {\rm Li}_2\left (e x\right ) - 2 \, b e^{3} n x^{3} - 3 \, b e^{2} n x^{2} - 6 \, b e n x + 6 \, {\left (b e^{3} n x^{3} - b n\right )} \log \left (-e x + 1\right )\right )} \log \relax (x) - 108 \, {\left (3 \, b e^{3} n x^{3} \log \relax (x) + 3 \, b e^{3} x^{3} \log \relax (c) - {\left (b e^{3} n - 3 \, a e^{3}\right )} x^{3}\right )} {\rm polylog}\left (3, e x\right )}{972 \, e^{3}} \]
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 \[ \int {\left (b \log \left (c x^{n}\right ) + a\right )} x^{2} {\rm Li}_{3}(e x)\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.43, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \,x^{n}\right )+a \right ) x^{2} \polylog \left (3, e x \right )\, dx \]
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 \[ -\frac {1}{162} \, b {\left (\frac {6 \, {\left (3 \, e^{3} x^{3} \log \left (x^{n}\right ) - {\left (2 \, e^{3} n - 3 \, e^{3} \log \relax (c)\right )} x^{3}\right )} {\rm Li}_2\left (e x\right ) - 6 \, {\left ({\left (e^{3} n - e^{3} \log \relax (c)\right )} x^{3} - n \log \relax (x)\right )} \log \left (-e x + 1\right ) - {\left (2 \, e^{3} x^{3} + 3 \, e^{2} x^{2} + 6 \, e x - 6 \, {\left (e^{3} x^{3} - 1\right )} \log \left (-e x + 1\right )\right )} \log \left (x^{n}\right ) - 18 \, {\left (3 \, e^{3} x^{3} \log \left (x^{n}\right ) - {\left (e^{3} n - 3 \, e^{3} \log \relax (c)\right )} x^{3}\right )} {\rm Li}_{3}(e x)}{e^{3}} - 162 \, \int -\frac {e^{2} n x^{2} + 2 \, {\left (4 \, e^{3} n - 3 \, e^{3} \log \relax (c)\right )} x^{3} + 3 \, e n x - 6 \, n \log \relax (x) - 6 \, n}{162 \, {\left (e^{3} x - e^{2}\right )}}\,{d x}\right )} - \frac {{\left (18 \, e^{3} x^{3} {\rm Li}_2\left (e x\right ) - 54 \, e^{3} x^{3} {\rm Li}_{3}(e x) - 2 \, e^{3} x^{3} - 3 \, e^{2} x^{2} - 6 \, e x + 6 \, {\left (e^{3} x^{3} - 1\right )} \log \left (-e x + 1\right )\right )} a}{162 \, e^{3}} \]
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 \[ \text {Hanged} \]
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 \[ \int x^{2} \left (a + b \log {\left (c x^{n} \right )}\right ) \operatorname {Li}_{3}\left (e x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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