Optimal. Leaf size=217 \[ -\frac {\log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{9 e^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{9 e^2}+\frac {1}{3} x^3 \text {Li}_2(e x) \left (a+b \log \left (c x^n\right )\right )+\frac {1}{9} x^3 \log (1-e x) \left (a+b \log \left (c x^n\right )\right )-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{18 e}-\frac {1}{27} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {b n \text {Li}_2(e x)}{9 e^3}+\frac {2 b n \log (1-e x)}{27 e^3}+\frac {5 b n x}{27 e^2}-\frac {1}{9} b n x^3 \text {Li}_2(e x)-\frac {2}{27} b n x^3 \log (1-e x)+\frac {7 b n x^2}{108 e}+\frac {1}{27} b n x^3 \]
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Rubi [A] time = 0.18, 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 = {2385, 2395, 43, 2376, 2391} \[ \frac {1}{3} x^3 \text {PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )-\frac {b n \text {PolyLog}(2,e x)}{9 e^3}-\frac {1}{9} b n x^3 \text {PolyLog}(2,e x)-\frac {x \left (a+b \log \left (c x^n\right )\right )}{9 e^2}-\frac {\log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{9 e^3}+\frac {1}{9} x^3 \log (1-e x) \left (a+b \log \left (c x^n\right )\right )-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{18 e}-\frac {1}{27} x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {5 b n x}{27 e^2}+\frac {2 b n \log (1-e x)}{27 e^3}+\frac {7 b n x^2}{108 e}-\frac {2}{27} b n x^3 \log (1-e x)+\frac {1}{27} b n x^3 \]
Antiderivative was successfully verified.
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Rule 43
Rule 2376
Rule 2385
Rule 2391
Rule 2395
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 = 0.55, size = 196, normalized size = 0.90 \[ \frac {\left (18 e^3 x^3 \text {Li}_2(e x)+6 \left (e^3 x^3-1\right ) \log (1-e x)-e x \left (2 e^2 x^2+3 e x+6\right )\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{54 e^3}+\frac {b n \left (12 \text {Li}_2(e x) \left (-e^3 x^3+3 e^3 x^3 \log (x)-1\right )+4 e^3 x^3-8 e^3 x^3 \log (1-e x)+7 e^2 x^2+2 \log (x) \left (6 \left (e^3 x^3-1\right ) \log (1-e x)-e x \left (2 e^2 x^2+3 e x+6\right )\right )+20 e x+8 \log (1-e x)\right )}{108 e^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 247, normalized size = 1.14 \[ \frac {4 \, {\left (b e^{3} n - a e^{3}\right )} x^{3} + {\left (7 \, b e^{2} n - 6 \, a e^{2}\right )} x^{2} + 4 \, {\left (5 \, b e n - 3 \, a e\right )} x - 12 \, {\left ({\left (b e^{3} n - 3 \, a e^{3}\right )} x^{3} + b n\right )} {\rm Li}_2\left (e x\right ) - 4 \, {\left ({\left (2 \, b e^{3} n - 3 \, a e^{3}\right )} x^{3} - 2 \, b n + 3 \, a\right )} \log \left (-e x + 1\right ) + 2 \, {\left (18 \, b e^{3} x^{3} {\rm Li}_2\left (e x\right ) - 2 \, b e^{3} x^{3} - 3 \, b e^{2} x^{2} - 6 \, b e x + 6 \, {\left (b e^{3} x^{3} - b\right )} \log \left (-e x + 1\right )\right )} \log \relax (c) + 2 \, {\left (18 \, b e^{3} n x^{3} {\rm Li}_2\left (e x\right ) - 2 \, b e^{3} n x^{3} - 3 \, b e^{2} n x^{2} - 6 \, b e n x + 6 \, {\left (b e^{3} n x^{3} - b n\right )} \log \left (-e x + 1\right )\right )} \log \relax (x)}{108 \, e^{3}} \]
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 \[ \int {\left (b \log \left (c x^{n}\right ) + a\right )} x^{2} {\rm Li}_2\left (e x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.28, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \,x^{n}\right )+a \right ) x^{2} \polylog \left (2, e x \right )\, dx \]
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 \[ \frac {1}{54} \, b {\left (\frac {6 \, {\left (3 \, e^{3} x^{3} \log \left (x^{n}\right ) - {\left (e^{3} n - 3 \, e^{3} \log \relax (c)\right )} x^{3}\right )} {\rm Li}_2\left (e x\right ) - 2 \, {\left ({\left (2 \, e^{3} n - 3 \, e^{3} \log \relax (c)\right )} x^{3} - 3 \, n \log \relax (x)\right )} \log \left (-e x + 1\right ) - {\left (2 \, e^{3} x^{3} + 3 \, e^{2} x^{2} + 6 \, e x - 6 \, {\left (e^{3} x^{3} - 1\right )} \log \left (-e x + 1\right )\right )} \log \left (x^{n}\right )}{e^{3}} - 54 \, \int -\frac {e^{2} n x^{2} + 6 \, {\left (e^{3} n - e^{3} \log \relax (c)\right )} x^{3} + 3 \, e n x - 6 \, n \log \relax (x) - 6 \, n}{54 \, {\left (e^{3} x - e^{2}\right )}}\,{d x}\right )} + \frac {{\left (18 \, e^{3} x^{3} {\rm Li}_2\left (e x\right ) - 2 \, e^{3} x^{3} - 3 \, e^{2} x^{2} - 6 \, e x + 6 \, {\left (e^{3} x^{3} - 1\right )} \log \left (-e x + 1\right )\right )} a}{54 \, e^{3}} \]
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 \[ \int x^2\,\mathrm {polylog}\left (2,e\,x\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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