Optimal. Leaf size=396 \[ \frac {2 b n \text {Li}_2(-e x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{3 e^3}-\frac {2 b n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{9 e^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2}+\frac {2 b n x \left (a+b \log \left (c x^n\right )\right )}{9 e^2}+\frac {1}{3} x^3 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2-\frac {2}{9} b n x^3 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{6 e}-\frac {5 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{18 e}-\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac {4}{27} b n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {2 a b n x}{3 e^2}+\frac {2 b^2 n x \log \left (c x^n\right )}{3 e^2}-\frac {2 b^2 n^2 \text {Li}_2(-e x)}{9 e^3}-\frac {2 b^2 n^2 \text {Li}_3(-e x)}{3 e^3}+\frac {2 b^2 n^2 \log (e x+1)}{27 e^3}-\frac {26 b^2 n^2 x}{27 e^2}+\frac {2}{27} b^2 n^2 x^3 \log (e x+1)+\frac {19 b^2 n^2 x^2}{108 e}-\frac {2}{27} b^2 n^2 x^3 \]
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Rubi [A] time = 0.29, antiderivative size = 396, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {2395, 43, 2377, 2295, 2304, 2374, 6589, 2376, 2391} \[ \frac {2 b n \text {PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac {2 b^2 n^2 \text {PolyLog}(2,-e x)}{9 e^3}-\frac {2 b^2 n^2 \text {PolyLog}(3,-e x)}{3 e^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2}+\frac {2 b n x \left (a+b \log \left (c x^n\right )\right )}{9 e^2}+\frac {\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{3 e^3}-\frac {2 b n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{9 e^3}+\frac {1}{3} x^3 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2-\frac {2}{9} b n x^3 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{6 e}-\frac {5 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{18 e}-\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac {4}{27} b n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {2 a b n x}{3 e^2}+\frac {2 b^2 n x \log \left (c x^n\right )}{3 e^2}-\frac {26 b^2 n^2 x}{27 e^2}+\frac {2 b^2 n^2 \log (e x+1)}{27 e^3}+\frac {19 b^2 n^2 x^2}{108 e}+\frac {2}{27} b^2 n^2 x^3 \log (e x+1)-\frac {2}{27} b^2 n^2 x^3 \]
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^2 \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}{3 e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{6 e}-\frac {1}{9} x^3 \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)}{3 e^3}+\frac {1}{3} x^3 \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 )}{3 e^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )}{6 e}-\frac {1}{9} x^2 \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 x}+\frac {1}{3} x^2 \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}{3 e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{6 e}-\frac {1}{9} x^3 \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)}{3 e^3}+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)+\frac {1}{9} (2 b n) \int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx-\frac {1}{3} (2 b n) \int x^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x) \, dx-\frac {(2 b n) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{x} \, dx}{3 e^3}+\frac {(2 b n) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{3 e^2}-\frac {(b n) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{3 e}\\ &=\frac {2 a b n x}{3 e^2}+\frac {b^2 n^2 x^2}{12 e}-\frac {2}{81} b^2 n^2 x^3+\frac {2 b n x \left (a+b \log \left (c x^n\right )\right )}{9 e^2}-\frac {5 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{18 e}+\frac {4}{27} b n x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{6 e}-\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{9 e^3}-\frac {2}{9} b n x^3 \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)}{3 e^3}+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{3 e^3}+\frac {\left (2 b^2 n\right ) \int \log \left (c x^n\right ) \, dx}{3 e^2}+\frac {1}{3} \left (2 b^2 n^2\right ) \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 {\left (2 b^2 n^2\right ) \int \frac {\text {Li}_2(-e x)}{x} \, dx}{3 e^3}\\ &=\frac {2 a b n x}{3 e^2}-\frac {8 b^2 n^2 x}{9 e^2}+\frac {5 b^2 n^2 x^2}{36 e}-\frac {4}{81} b^2 n^2 x^3+\frac {2 b^2 n x \log \left (c x^n\right )}{3 e^2}+\frac {2 b n x \left (a+b \log \left (c x^n\right )\right )}{9 e^2}-\frac {5 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{18 e}+\frac {4}{27} b n x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{6 e}-\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{9 e^3}-\frac {2}{9} b n x^3 \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)}{3 e^3}+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{3 e^3}-\frac {2 b^2 n^2 \text {Li}_3(-e x)}{3 e^3}+\frac {1}{9} \left (2 b^2 n^2\right ) \int x^2 \log (1+e x) \, dx+\frac {\left (2 b^2 n^2\right ) \int \frac {\log (1+e x)}{x} \, dx}{9 e^3}\\ &=\frac {2 a b n x}{3 e^2}-\frac {8 b^2 n^2 x}{9 e^2}+\frac {5 b^2 n^2 x^2}{36 e}-\frac {4}{81} b^2 n^2 x^3+\frac {2 b^2 n x \log \left (c x^n\right )}{3 e^2}+\frac {2 b n x \left (a+b \log \left (c x^n\right )\right )}{9 e^2}-\frac {5 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{18 e}+\frac {4}{27} b n x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{6 e}-\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac {2}{27} b^2 n^2 x^3 \log (1+e x)-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{9 e^3}-\frac {2}{9} b n x^3 \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)}{3 e^3}+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)-\frac {2 b^2 n^2 \text {Li}_2(-e x)}{9 e^3}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{3 e^3}-\frac {2 b^2 n^2 \text {Li}_3(-e x)}{3 e^3}-\frac {1}{27} \left (2 b^2 e n^2\right ) \int \frac {x^3}{1+e x} \, dx\\ &=\frac {2 a b n x}{3 e^2}-\frac {8 b^2 n^2 x}{9 e^2}+\frac {5 b^2 n^2 x^2}{36 e}-\frac {4}{81} b^2 n^2 x^3+\frac {2 b^2 n x \log \left (c x^n\right )}{3 e^2}+\frac {2 b n x \left (a+b \log \left (c x^n\right )\right )}{9 e^2}-\frac {5 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{18 e}+\frac {4}{27} b n x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{6 e}-\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac {2}{27} b^2 n^2 x^3 \log (1+e x)-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{9 e^3}-\frac {2}{9} b n x^3 \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)}{3 e^3}+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)-\frac {2 b^2 n^2 \text {Li}_2(-e x)}{9 e^3}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{3 e^3}-\frac {2 b^2 n^2 \text {Li}_3(-e x)}{3 e^3}-\frac {1}{27} \left (2 b^2 e n^2\right ) \int \left (\frac {1}{e^3}-\frac {x}{e^2}+\frac {x^2}{e}-\frac {1}{e^3 (1+e x)}\right ) \, dx\\ &=\frac {2 a b n x}{3 e^2}-\frac {26 b^2 n^2 x}{27 e^2}+\frac {19 b^2 n^2 x^2}{108 e}-\frac {2}{27} b^2 n^2 x^3+\frac {2 b^2 n x \log \left (c x^n\right )}{3 e^2}+\frac {2 b n x \left (a+b \log \left (c x^n\right )\right )}{9 e^2}-\frac {5 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{18 e}+\frac {4}{27} b n x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{6 e}-\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac {2 b^2 n^2 \log (1+e x)}{27 e^3}+\frac {2}{27} b^2 n^2 x^3 \log (1+e x)-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{9 e^3}-\frac {2}{9} b n x^3 \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)}{3 e^3}+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)-\frac {2 b^2 n^2 \text {Li}_2(-e x)}{9 e^3}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{3 e^3}-\frac {2 b^2 n^2 \text {Li}_3(-e x)}{3 e^3}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 506, normalized size = 1.28 \[ \frac {-12 a^2 e^3 x^3+36 a^2 e^3 x^3 \log (e x+1)+18 a^2 e^2 x^2-36 a^2 e x+36 a^2 \log (e x+1)-24 a b e^3 x^3 \log \left (c x^n\right )+72 a b e^3 x^3 \log (e x+1) \log \left (c x^n\right )+36 a b e^2 x^2 \log \left (c x^n\right )+24 b n \text {Li}_2(-e x) \left (3 a+3 b \log \left (c x^n\right )-b n\right )-72 a b e x \log \left (c x^n\right )+72 a b \log (e x+1) \log \left (c x^n\right )+16 a b e^3 n x^3-24 a b e^3 n x^3 \log (e x+1)-30 a b e^2 n x^2+96 a b e n x-24 a b n \log (e x+1)-12 b^2 e^3 x^3 \log ^2\left (c x^n\right )+36 b^2 e^3 x^3 \log (e x+1) \log ^2\left (c x^n\right )+16 b^2 e^3 n x^3 \log \left (c x^n\right )-24 b^2 e^3 n x^3 \log (e x+1) \log \left (c x^n\right )+18 b^2 e^2 x^2 \log ^2\left (c x^n\right )-30 b^2 e^2 n x^2 \log \left (c x^n\right )-36 b^2 e x \log ^2\left (c x^n\right )+36 b^2 \log (e x+1) \log ^2\left (c x^n\right )+96 b^2 e n x \log \left (c x^n\right )-24 b^2 n \log (e x+1) \log \left (c x^n\right )-8 b^2 e^3 n^2 x^3+8 b^2 e^3 n^2 x^3 \log (e x+1)+19 b^2 e^2 n^2 x^2-72 b^2 n^2 \text {Li}_3(-e x)-104 b^2 e n^2 x+8 b^2 n^2 \log (e x+1)}{108 e^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{2} x^{2} \log \left (c x^{n}\right )^{2} \log \left (e x + 1\right ) + 2 \, a b x^{2} \log \left (c x^{n}\right ) \log \left (e x + 1\right ) + a^{2} x^{2} \log \left (e x + 1\right ), x\right ) \]
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 \[ \int {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x^{2} \log \left (e x + 1\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.51, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \,x^{n}\right )+a \right )^{2} x^{2} \ln \left (e x +1\right )\, dx \]
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 \[ -\frac {{\left (2 \, b^{2} e^{3} x^{3} - 3 \, b^{2} e^{2} x^{2} + 6 \, b^{2} e x - 6 \, {\left (b^{2} e^{3} x^{3} + b^{2}\right )} \log \left (e x + 1\right )\right )} \log \left (x^{n}\right )^{2}}{18 \, e^{3}} + \frac {-\frac {2}{9} \, b^{2} e^{3} n^{2} x^{3} + \frac {2}{3} \, b^{2} e^{3} n x^{3} \log \left (x^{n}\right ) + \frac {3}{4} \, b^{2} e^{2} n^{2} x^{2} + \frac {1}{2} \, {\left (6 \, x^{3} \log \left (e x + 1\right ) - e {\left (\frac {2 \, e^{2} x^{3} - 3 \, e x^{2} + 6 \, x}{e^{3}} - \frac {6 \, \log \left (e x + 1\right )}{e^{4}}\right )}\right )} b^{2} e^{3} \log \relax (c)^{2} - \frac {3}{2} \, b^{2} e^{2} n x^{2} \log \left (x^{n}\right ) + {\left (6 \, x^{3} \log \left (e x + 1\right ) - e {\left (\frac {2 \, e^{2} x^{3} - 3 \, e x^{2} + 6 \, x}{e^{3}} - \frac {6 \, \log \left (e x + 1\right )}{e^{4}}\right )}\right )} a b e^{3} \log \relax (c) + \frac {1}{2} \, {\left (6 \, x^{3} \log \left (e x + 1\right ) - e {\left (\frac {2 \, e^{2} x^{3} - 3 \, e x^{2} + 6 \, x}{e^{3}} - \frac {6 \, \log \left (e x + 1\right )}{e^{4}}\right )}\right )} a^{2} e^{3} - 6 \, b^{2} e n^{2} x + 6 \, b^{2} e n x \log \left (x^{n}\right ) + \int \frac {6 \, {\left ({\left (3 \, a b e^{3} - {\left (e^{3} n - 3 \, e^{3} \log \relax (c)\right )} b^{2}\right )} x^{3} - b^{2} n\right )} \log \left (e x + 1\right ) \log \left (x^{n}\right )}{x}\,{d x}}{9 \, e^{3}} \]
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 \[ \int x^2\,\ln \left (e\,x+1\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2 \,d x \]
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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