Optimal. Leaf size=327 \[ -\frac {b n \text {Li}_2(-e x) \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {b n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac {\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2}-\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{2 e}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{2 e}-\frac {1}{2} b n x^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )+\frac {1}{2} x^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {a b n x}{e}-\frac {b^2 n x \log \left (c x^n\right )}{e}+\frac {b^2 n^2 \text {Li}_2(-e x)}{2 e^2}+\frac {b^2 n^2 \text {Li}_3(-e x)}{e^2}-\frac {b^2 n^2 \log (e x+1)}{4 e^2}+\frac {1}{4} b^2 n^2 x^2 \log (e x+1)+\frac {7 b^2 n^2 x}{4 e}-\frac {3}{8} b^2 n^2 x^2 \]
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Rubi [A] time = 0.22, antiderivative size = 327, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {2395, 43, 2377, 2295, 2304, 2374, 6589, 2376, 2391} \[ -\frac {b n \text {PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {b^2 n^2 \text {PolyLog}(2,-e x)}{2 e^2}+\frac {b^2 n^2 \text {PolyLog}(3,-e x)}{e^2}+\frac {b n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac {\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2}-\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{2 e}-\frac {1}{2} b n x^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{2 e}+\frac {1}{2} x^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {a b n x}{e}-\frac {b^2 n x \log \left (c x^n\right )}{e}-\frac {b^2 n^2 \log (e x+1)}{4 e^2}+\frac {1}{4} b^2 n^2 x^2 \log (e x+1)+\frac {7 b^2 n^2 x}{4 e}-\frac {3}{8} b^2 n^2 x^2 \]
Antiderivative was successfully verified.
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Rule 43
Rule 2295
Rule 2304
Rule 2374
Rule 2376
Rule 2377
Rule 2391
Rule 2395
Rule 6589
Rubi steps
\begin {align*} \int x \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x) \, dx &=\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{2 e}-\frac {1}{4} x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{2 e^2}+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)-(2 b n) \int \left (\frac {a+b \log \left (c x^n\right )}{2 e}-\frac {1}{4} x \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)}{2 e^2 x}+\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)\right ) \, dx\\ &=\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{2 e}-\frac {1}{4} x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{2 e^2}+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)+\frac {1}{2} (b n) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx-(b n) \int x \left (a+b \log \left (c x^n\right )\right ) \log (1+e x) \, dx+\frac {(b n) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{x} \, dx}{e^2}-\frac {(b n) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{e}\\ &=-\frac {a b n x}{e}-\frac {1}{8} b^2 n^2 x^2-\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{2 e}+\frac {1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{2 e}-\frac {1}{4} x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{2 e^2}-\frac {1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{2 e^2}+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{e^2}-\frac {\left (b^2 n\right ) \int \log \left (c x^n\right ) \, dx}{e}+\left (b^2 n^2\right ) \int \left (\frac {1}{2 e}-\frac {x}{4}-\frac {\log (1+e x)}{2 e^2 x}+\frac {1}{2} x \log (1+e x)\right ) \, dx+\frac {\left (b^2 n^2\right ) \int \frac {\text {Li}_2(-e x)}{x} \, dx}{e^2}\\ &=-\frac {a b n x}{e}+\frac {3 b^2 n^2 x}{2 e}-\frac {1}{4} b^2 n^2 x^2-\frac {b^2 n x \log \left (c x^n\right )}{e}-\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{2 e}+\frac {1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{2 e}-\frac {1}{4} x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{2 e^2}-\frac {1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{2 e^2}+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{e^2}+\frac {b^2 n^2 \text {Li}_3(-e x)}{e^2}+\frac {1}{2} \left (b^2 n^2\right ) \int x \log (1+e x) \, dx-\frac {\left (b^2 n^2\right ) \int \frac {\log (1+e x)}{x} \, dx}{2 e^2}\\ &=-\frac {a b n x}{e}+\frac {3 b^2 n^2 x}{2 e}-\frac {1}{4} b^2 n^2 x^2-\frac {b^2 n x \log \left (c x^n\right )}{e}-\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{2 e}+\frac {1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{2 e}-\frac {1}{4} x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{4} b^2 n^2 x^2 \log (1+e x)+\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{2 e^2}-\frac {1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{2 e^2}+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)+\frac {b^2 n^2 \text {Li}_2(-e x)}{2 e^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{e^2}+\frac {b^2 n^2 \text {Li}_3(-e x)}{e^2}-\frac {1}{4} \left (b^2 e n^2\right ) \int \frac {x^2}{1+e x} \, dx\\ &=-\frac {a b n x}{e}+\frac {3 b^2 n^2 x}{2 e}-\frac {1}{4} b^2 n^2 x^2-\frac {b^2 n x \log \left (c x^n\right )}{e}-\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{2 e}+\frac {1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{2 e}-\frac {1}{4} x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{4} b^2 n^2 x^2 \log (1+e x)+\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{2 e^2}-\frac {1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{2 e^2}+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)+\frac {b^2 n^2 \text {Li}_2(-e x)}{2 e^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{e^2}+\frac {b^2 n^2 \text {Li}_3(-e x)}{e^2}-\frac {1}{4} \left (b^2 e n^2\right ) \int \left (-\frac {1}{e^2}+\frac {x}{e}+\frac {1}{e^2 (1+e x)}\right ) \, dx\\ &=-\frac {a b n x}{e}+\frac {7 b^2 n^2 x}{4 e}-\frac {3}{8} b^2 n^2 x^2-\frac {b^2 n x \log \left (c x^n\right )}{e}-\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{2 e}+\frac {1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{2 e}-\frac {1}{4} x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {b^2 n^2 \log (1+e x)}{4 e^2}+\frac {1}{4} b^2 n^2 x^2 \log (1+e x)+\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{2 e^2}-\frac {1}{2} b n x^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{2 e^2}+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)+\frac {b^2 n^2 \text {Li}_2(-e x)}{2 e^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{e^2}+\frac {b^2 n^2 \text {Li}_3(-e x)}{e^2}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 416, normalized size = 1.27 \[ \frac {-2 a^2 e^2 x^2+4 a^2 e^2 x^2 \log (e x+1)+4 a^2 e x-4 a^2 \log (e x+1)-4 a b e^2 x^2 \log \left (c x^n\right )+8 a b e^2 x^2 \log (e x+1) \log \left (c x^n\right )+4 b n \text {Li}_2(-e x) \left (-2 a-2 b \log \left (c x^n\right )+b n\right )+8 a b e x \log \left (c x^n\right )-8 a b \log (e x+1) \log \left (c x^n\right )+4 a b e^2 n x^2-4 a b e^2 n x^2 \log (e x+1)-12 a b e n x+4 a b n \log (e x+1)-2 b^2 e^2 x^2 \log ^2\left (c x^n\right )+4 b^2 e^2 x^2 \log (e x+1) \log ^2\left (c x^n\right )+4 b^2 e^2 n x^2 \log \left (c x^n\right )-4 b^2 e^2 n x^2 \log (e x+1) \log \left (c x^n\right )+4 b^2 e x \log ^2\left (c x^n\right )-4 b^2 \log (e x+1) \log ^2\left (c x^n\right )-12 b^2 e n x \log \left (c x^n\right )+4 b^2 n \log (e x+1) \log \left (c x^n\right )-3 b^2 e^2 n^2 x^2+2 b^2 e^2 n^2 x^2 \log (e x+1)+8 b^2 n^2 \text {Li}_3(-e x)+14 b^2 e n^2 x-2 b^2 n^2 \log (e x+1)}{8 e^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.84, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{2} x \log \left (c x^{n}\right )^{2} \log \left (e x + 1\right ) + 2 \, a b x \log \left (c x^{n}\right ) \log \left (e x + 1\right ) + a^{2} x \log \left (e x + 1\right ), x\right ) \]
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 \[ \int {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x \log \left (e x + 1\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.66, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \,x^{n}\right )+a \right )^{2} x \ln \left (e x +1\right )\, dx \]
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 \[ -\frac {{\left (b^{2} e^{2} x^{2} - 2 \, b^{2} e x - 2 \, {\left (b^{2} e^{2} x^{2} - b^{2}\right )} \log \left (e x + 1\right )\right )} \log \left (x^{n}\right )^{2}}{4 \, e^{2}} + \frac {-\frac {1}{4} \, b^{2} e^{2} n^{2} x^{2} + \frac {1}{2} \, b^{2} e^{2} n x^{2} \log \left (x^{n}\right ) + \frac {1}{2} \, {\left (2 \, x^{2} \log \left (e x + 1\right ) - e {\left (\frac {e x^{2} - 2 \, x}{e^{2}} + \frac {2 \, \log \left (e x + 1\right )}{e^{3}}\right )}\right )} b^{2} e^{2} \log \relax (c)^{2} + 2 \, b^{2} e n^{2} x + {\left (2 \, x^{2} \log \left (e x + 1\right ) - e {\left (\frac {e x^{2} - 2 \, x}{e^{2}} + \frac {2 \, \log \left (e x + 1\right )}{e^{3}}\right )}\right )} a b e^{2} \log \relax (c) - 2 \, b^{2} e n x \log \left (x^{n}\right ) + \frac {1}{2} \, {\left (2 \, x^{2} \log \left (e x + 1\right ) - e {\left (\frac {e x^{2} - 2 \, x}{e^{2}} + \frac {2 \, \log \left (e x + 1\right )}{e^{3}}\right )}\right )} a^{2} e^{2} + \int \frac {2 \, {\left (b^{2} n + {\left (2 \, a b e^{2} - {\left (e^{2} n - 2 \, e^{2} \log \relax (c)\right )} b^{2}\right )} x^{2}\right )} \log \left (e x + 1\right ) \log \left (x^{n}\right )}{x}\,{d x}}{2 \, e^{2}} \]
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 \[ \int x\,\ln \left (e\,x+1\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2 \,d x \]
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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