Optimal. Leaf size=178 \[ \frac {\log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{3 e^2}+\frac {1}{3} x^3 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{6 e}-\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {b n \text {Li}_2(-e x)}{3 e^3}-\frac {b n \log (e x+1)}{9 e^3}+\frac {4 b n x}{9 e^2}-\frac {1}{9} b n x^3 \log (e x+1)-\frac {5 b n x^2}{36 e}+\frac {2}{27} b n x^3 \]
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Rubi [A] time = 0.10, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2395, 43, 2376, 2391} \[ \frac {b n \text {PolyLog}(2,-e x)}{3 e^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{3 e^2}+\frac {\log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {1}{3} x^3 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{6 e}-\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {4 b n x}{9 e^2}-\frac {b n \log (e x+1)}{9 e^3}-\frac {5 b n x^2}{36 e}-\frac {1}{9} b n x^3 \log (e x+1)+\frac {2}{27} b n x^3 \]
Antiderivative was successfully verified.
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Rule 43
Rule 2376
Rule 2391
Rule 2395
Rubi steps
\begin {align*} \int x^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x) \, dx &=-\frac {x \left (a+b \log \left (c x^n\right )\right )}{3 e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{6 e}-\frac {1}{9} 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)}{3 e^3}+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-(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 {b n x}{3 e^2}-\frac {b n x^2}{12 e}+\frac {1}{27} b n x^3-\frac {x \left (a+b \log \left (c x^n\right )\right )}{3 e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{6 e}-\frac {1}{9} 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)}{3 e^3}+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac {1}{3} (b n) \int x^2 \log (1+e x) \, dx-\frac {(b n) \int \frac {\log (1+e x)}{x} \, dx}{3 e^3}\\ &=\frac {b n x}{3 e^2}-\frac {b n x^2}{12 e}+\frac {1}{27} b n x^3-\frac {x \left (a+b \log \left (c x^n\right )\right )}{3 e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{6 e}-\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{9} b n x^3 \log (1+e x)+\frac {\left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{3 e^3}+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)+\frac {b n \text {Li}_2(-e x)}{3 e^3}+\frac {1}{9} (b e n) \int \frac {x^3}{1+e x} \, dx\\ &=\frac {b n x}{3 e^2}-\frac {b n x^2}{12 e}+\frac {1}{27} b n x^3-\frac {x \left (a+b \log \left (c x^n\right )\right )}{3 e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{6 e}-\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{9} b n x^3 \log (1+e x)+\frac {\left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{3 e^3}+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)+\frac {b n \text {Li}_2(-e x)}{3 e^3}+\frac {1}{9} (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}{9 e^2}-\frac {5 b n x^2}{36 e}+\frac {2}{27} b n x^3-\frac {x \left (a+b \log \left (c x^n\right )\right )}{3 e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{6 e}-\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {b n \log (1+e x)}{9 e^3}-\frac {1}{9} b n x^3 \log (1+e x)+\frac {\left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{3 e^3}+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)+\frac {b n \text {Li}_2(-e x)}{3 e^3}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 161, normalized size = 0.90 \[ \frac {-12 a e^3 x^3+36 a e^3 x^3 \log (e x+1)+18 a e^2 x^2-36 a e x+36 a \log (e x+1)+6 b \left (6 \left (e^3 x^3+1\right ) \log (e x+1)+e x \left (-2 e^2 x^2+3 e x-6\right )\right ) \log \left (c x^n\right )+8 b e^3 n x^3-12 b e^3 n x^3 \log (e x+1)-15 b e^2 n x^2+36 b n \text {Li}_2(-e x)+48 b e n x-12 b n \log (e x+1)}{108 e^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b x^{2} \log \left (c x^{n}\right ) \log \left (e x + 1\right ) + a x^{2} \log \left (e x + 1\right ), x\right ) \]
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 \[ \int {\left (b \log \left (c x^{n}\right ) + a\right )} x^{2} \log \left (e x + 1\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.29, size = 870, normalized size = 4.89 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 220, normalized size = 1.24 \[ \frac {{\left (\log \left (e x + 1\right ) \log \relax (x) + {\rm Li}_2\left (-e x\right )\right )} b n}{3 \, e^{3}} - \frac {{\left (b {\left (n - 3 \, \log \relax (c)\right )} - 3 \, a\right )} \log \left (e x + 1\right )}{9 \, e^{3}} - \frac {4 \, {\left (3 \, a e^{3} - {\left (2 \, e^{3} n - 3 \, e^{3} \log \relax (c)\right )} b\right )} x^{3} - 3 \, {\left (6 \, a e^{2} - {\left (5 \, e^{2} n - 6 \, e^{2} \log \relax (c)\right )} b\right )} x^{2} - 12 \, {\left ({\left (4 \, e n - 3 \, e \log \relax (c)\right )} b - 3 \, a e\right )} x - 12 \, {\left ({\left (3 \, a e^{3} - {\left (e^{3} n - 3 \, e^{3} \log \relax (c)\right )} b\right )} x^{3} - 3 \, b n \log \relax (x)\right )} \log \left (e x + 1\right ) + 6 \, {\left (2 \, b e^{3} x^{3} - 3 \, b e^{2} x^{2} + 6 \, b e x - 6 \, {\left (b e^{3} x^{3} + b\right )} \log \left (e x + 1\right )\right )} \log \left (x^{n}\right )}{108 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,\ln \left (e\,x+1\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]
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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