Optimal. Leaf size=615 \[ -\frac {2 b^2 n^2 \text {Li}_2(-e x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac {2 b^2 n^2 \text {Li}_3(-e x) \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac {2 b^2 n^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{9 e^3}-\frac {2 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{9 e^2}+\frac {2}{9} b^2 n^2 x^3 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )+\frac {19 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{36 e}-\frac {2}{9} b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {8 a b^2 n^2 x}{3 e^2}+\frac {b n \text {Li}_2(-e x) \left (a+b \log \left (c x^n\right )\right )^2}{e^3}-\frac {b n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{3 e^3}+\frac {\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3}{3 e^3}+\frac {4 b n x \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2}-\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{3 e^2}-\frac {1}{3} b n x^3 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{3} x^3 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3-\frac {5 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 e}+\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )^3-\frac {8 b^3 n^2 x \log \left (c x^n\right )}{3 e^2}+\frac {2 b^3 n^3 \text {Li}_2(-e x)}{9 e^3}+\frac {2 b^3 n^3 \text {Li}_3(-e x)}{3 e^3}+\frac {2 b^3 n^3 \text {Li}_4(-e x)}{e^3}-\frac {2 b^3 n^3 \log (e x+1)}{27 e^3}+\frac {80 b^3 n^3 x}{27 e^2}-\frac {2}{27} b^3 n^3 x^3 \log (e x+1)-\frac {65 b^3 n^3 x^2}{216 e}+\frac {8}{81} b^3 n^3 x^3 \]
[Out]
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Rubi [A] time = 0.64, antiderivative size = 615, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 12, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.546, Rules used = {2395, 43, 2377, 2296, 2295, 2305, 2304, 2374, 2383, 6589, 2376, 2391} \[ -\frac {2 b^2 n^2 \text {PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac {2 b^2 n^2 \text {PolyLog}(3,-e x) \left (a+b \log \left (c x^n\right )\right )}{e^3}+\frac {b n \text {PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )^2}{e^3}+\frac {2 b^3 n^3 \text {PolyLog}(2,-e x)}{9 e^3}+\frac {2 b^3 n^3 \text {PolyLog}(3,-e x)}{3 e^3}+\frac {2 b^3 n^3 \text {PolyLog}(4,-e x)}{e^3}-\frac {2 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{9 e^2}+\frac {2 b^2 n^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{9 e^3}+\frac {19 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{36 e}+\frac {2}{9} b^2 n^2 x^3 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )-\frac {2}{9} b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {8 a b^2 n^2 x}{3 e^2}+\frac {4 b n x \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2}-\frac {b n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{3 e^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{3 e^2}+\frac {\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3}{3 e^3}-\frac {5 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}-\frac {1}{3} b n x^3 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 e}+\frac {1}{3} x^3 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3+\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )^3-\frac {8 b^3 n^2 x \log \left (c x^n\right )}{3 e^2}+\frac {80 b^3 n^3 x}{27 e^2}-\frac {2 b^3 n^3 \log (e x+1)}{27 e^3}-\frac {65 b^3 n^3 x^2}{216 e}-\frac {2}{27} b^3 n^3 x^3 \log (e x+1)+\frac {8}{81} b^3 n^3 x^3 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 2295
Rule 2296
Rule 2304
Rule 2305
Rule 2374
Rule 2376
Rule 2377
Rule 2383
Rule 2391
Rule 2395
Rule 6589
Rubi steps
\begin {align*} \int x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x) \, dx &=-\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{3 e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 e}-\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )^3+\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{3 e^3}+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)-(3 b n) \int \left (-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 e^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{6 e}-\frac {1}{9} 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)}{3 e^3 x}+\frac {1}{3} x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)\right ) \, dx\\ &=-\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{3 e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 e}-\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )^3+\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{3 e^3}+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)+\frac {1}{3} (b n) \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx-(b n) \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x) \, dx-\frac {(b n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x} \, dx}{e^3}+\frac {(b n) \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{e^2}-\frac {(b n) \int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{2 e}\\ &=\frac {4 b n x \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2}-\frac {5 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}+\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{3 e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 e}-\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )^3-\frac {b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{3 e^3}-\frac {1}{3} b n x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)+\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{3 e^3}+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)+\frac {b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2(-e x)}{e^3}-\frac {1}{9} \left (2 b^2 n^2\right ) \int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx+\left (2 b^2 n^2\right ) \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 {\left (2 b^2 n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{x} \, dx}{e^3}-\frac {\left (2 b^2 n^2\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{e^2}+\frac {\left (b^2 n^2\right ) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{2 e}\\ &=-\frac {2 a b^2 n^2 x}{e^2}-\frac {b^3 n^3 x^2}{8 e}+\frac {2}{81} b^3 n^3 x^3+\frac {b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{4 e}-\frac {2}{27} b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {4 b n x \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2}-\frac {5 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}+\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{3 e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 e}-\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )^3-\frac {b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{3 e^3}-\frac {1}{3} b n x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)+\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{3 e^3}+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)+\frac {b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2(-e x)}{e^3}-\frac {2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(-e x)}{e^3}-\frac {1}{9} \left (2 b^2 n^2\right ) \int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx+\frac {1}{3} \left (2 b^2 n^2\right ) \int x^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x) \, dx+\frac {\left (2 b^2 n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{x} \, dx}{3 e^3}-\frac {\left (2 b^2 n^2\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{3 e^2}-\frac {\left (2 b^3 n^2\right ) \int \log \left (c x^n\right ) \, dx}{e^2}+\frac {\left (b^2 n^2\right ) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{3 e}+\frac {\left (2 b^3 n^3\right ) \int \frac {\text {Li}_3(-e x)}{x} \, dx}{e^3}\\ &=-\frac {8 a b^2 n^2 x}{3 e^2}+\frac {2 b^3 n^3 x}{e^2}-\frac {5 b^3 n^3 x^2}{24 e}+\frac {4}{81} b^3 n^3 x^3-\frac {2 b^3 n^2 x \log \left (c x^n\right )}{e^2}-\frac {2 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{9 e^2}+\frac {19 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{36 e}-\frac {2}{9} b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {4 b n x \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2}-\frac {5 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}+\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{3 e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 e}-\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )^3+\frac {2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{9 e^3}+\frac {2}{9} b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac {b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{3 e^3}-\frac {1}{3} b n x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)+\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{3 e^3}+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)-\frac {2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{3 e^3}+\frac {b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2(-e x)}{e^3}-\frac {2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(-e x)}{e^3}+\frac {2 b^3 n^3 \text {Li}_4(-e x)}{e^3}-\frac {\left (2 b^3 n^2\right ) \int \log \left (c x^n\right ) \, dx}{3 e^2}-\frac {1}{3} \left (2 b^3 n^3\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^3 n^3\right ) \int \frac {\text {Li}_2(-e x)}{x} \, dx}{3 e^3}\\ &=-\frac {8 a b^2 n^2 x}{3 e^2}+\frac {26 b^3 n^3 x}{9 e^2}-\frac {19 b^3 n^3 x^2}{72 e}+\frac {2}{27} b^3 n^3 x^3-\frac {8 b^3 n^2 x \log \left (c x^n\right )}{3 e^2}-\frac {2 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{9 e^2}+\frac {19 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{36 e}-\frac {2}{9} b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {4 b n x \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2}-\frac {5 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}+\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{3 e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 e}-\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )^3+\frac {2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{9 e^3}+\frac {2}{9} b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac {b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{3 e^3}-\frac {1}{3} b n x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)+\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{3 e^3}+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)-\frac {2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{3 e^3}+\frac {b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2(-e x)}{e^3}+\frac {2 b^3 n^3 \text {Li}_3(-e x)}{3 e^3}-\frac {2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(-e x)}{e^3}+\frac {2 b^3 n^3 \text {Li}_4(-e x)}{e^3}-\frac {1}{9} \left (2 b^3 n^3\right ) \int x^2 \log (1+e x) \, dx-\frac {\left (2 b^3 n^3\right ) \int \frac {\log (1+e x)}{x} \, dx}{9 e^3}\\ &=-\frac {8 a b^2 n^2 x}{3 e^2}+\frac {26 b^3 n^3 x}{9 e^2}-\frac {19 b^3 n^3 x^2}{72 e}+\frac {2}{27} b^3 n^3 x^3-\frac {8 b^3 n^2 x \log \left (c x^n\right )}{3 e^2}-\frac {2 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{9 e^2}+\frac {19 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{36 e}-\frac {2}{9} b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {4 b n x \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2}-\frac {5 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}+\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{3 e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 e}-\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )^3-\frac {2}{27} b^3 n^3 x^3 \log (1+e x)+\frac {2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{9 e^3}+\frac {2}{9} b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac {b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{3 e^3}-\frac {1}{3} b n x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)+\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{3 e^3}+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)+\frac {2 b^3 n^3 \text {Li}_2(-e x)}{9 e^3}-\frac {2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{3 e^3}+\frac {b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2(-e x)}{e^3}+\frac {2 b^3 n^3 \text {Li}_3(-e x)}{3 e^3}-\frac {2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(-e x)}{e^3}+\frac {2 b^3 n^3 \text {Li}_4(-e x)}{e^3}+\frac {1}{27} \left (2 b^3 e n^3\right ) \int \frac {x^3}{1+e x} \, dx\\ &=-\frac {8 a b^2 n^2 x}{3 e^2}+\frac {26 b^3 n^3 x}{9 e^2}-\frac {19 b^3 n^3 x^2}{72 e}+\frac {2}{27} b^3 n^3 x^3-\frac {8 b^3 n^2 x \log \left (c x^n\right )}{3 e^2}-\frac {2 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{9 e^2}+\frac {19 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{36 e}-\frac {2}{9} b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {4 b n x \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2}-\frac {5 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}+\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{3 e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 e}-\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )^3-\frac {2}{27} b^3 n^3 x^3 \log (1+e x)+\frac {2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{9 e^3}+\frac {2}{9} b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac {b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{3 e^3}-\frac {1}{3} b n x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)+\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{3 e^3}+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)+\frac {2 b^3 n^3 \text {Li}_2(-e x)}{9 e^3}-\frac {2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{3 e^3}+\frac {b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2(-e x)}{e^3}+\frac {2 b^3 n^3 \text {Li}_3(-e x)}{3 e^3}-\frac {2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(-e x)}{e^3}+\frac {2 b^3 n^3 \text {Li}_4(-e x)}{e^3}+\frac {1}{27} \left (2 b^3 e n^3\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 {8 a b^2 n^2 x}{3 e^2}+\frac {80 b^3 n^3 x}{27 e^2}-\frac {65 b^3 n^3 x^2}{216 e}+\frac {8}{81} b^3 n^3 x^3-\frac {8 b^3 n^2 x \log \left (c x^n\right )}{3 e^2}-\frac {2 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{9 e^2}+\frac {19 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{36 e}-\frac {2}{9} b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {4 b n x \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2}-\frac {5 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}+\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{3 e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 e}-\frac {1}{9} x^3 \left (a+b \log \left (c x^n\right )\right )^3-\frac {2 b^3 n^3 \log (1+e x)}{27 e^3}-\frac {2}{27} b^3 n^3 x^3 \log (1+e x)+\frac {2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{9 e^3}+\frac {2}{9} b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac {b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{3 e^3}-\frac {1}{3} b n x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)+\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{3 e^3}+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)+\frac {2 b^3 n^3 \text {Li}_2(-e x)}{9 e^3}-\frac {2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{3 e^3}+\frac {b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2(-e x)}{e^3}+\frac {2 b^3 n^3 \text {Li}_3(-e x)}{3 e^3}-\frac {2 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(-e x)}{e^3}+\frac {2 b^3 n^3 \text {Li}_4(-e x)}{e^3}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 975, normalized size = 1.59 \[ \frac {-72 e^3 x^3 a^3+108 e^2 x^2 a^3-216 e x a^3+216 e^3 x^3 \log (e x+1) a^3+216 \log (e x+1) a^3+144 b e^3 n x^3 a^2-270 b e^2 n x^2 a^2+864 b e n x a^2-216 b e^3 x^3 \log \left (c x^n\right ) a^2+324 b e^2 x^2 \log \left (c x^n\right ) a^2-648 b e x \log \left (c x^n\right ) a^2-216 b e^3 n x^3 \log (e x+1) a^2-216 b n \log (e x+1) a^2+648 b e^3 x^3 \log \left (c x^n\right ) \log (e x+1) a^2+648 b \log \left (c x^n\right ) \log (e x+1) a^2-144 b^2 e^3 n^2 x^3 a+342 b^2 e^2 n^2 x^2 a-216 b^2 e^3 x^3 \log ^2\left (c x^n\right ) a+324 b^2 e^2 x^2 \log ^2\left (c x^n\right ) a-648 b^2 e x \log ^2\left (c x^n\right ) a-1872 b^2 e n^2 x a+288 b^2 e^3 n x^3 \log \left (c x^n\right ) a-540 b^2 e^2 n x^2 \log \left (c x^n\right ) a+1728 b^2 e n x \log \left (c x^n\right ) a+144 b^2 e^3 n^2 x^3 \log (e x+1) a+144 b^2 n^2 \log (e x+1) a+648 b^2 e^3 x^3 \log ^2\left (c x^n\right ) \log (e x+1) a+648 b^2 \log ^2\left (c x^n\right ) \log (e x+1) a-432 b^2 e^3 n x^3 \log \left (c x^n\right ) \log (e x+1) a-432 b^2 n \log \left (c x^n\right ) \log (e x+1) a+64 b^3 e^3 n^3 x^3-72 b^3 e^3 x^3 \log ^3\left (c x^n\right )+108 b^3 e^2 x^2 \log ^3\left (c x^n\right )-216 b^3 e x \log ^3\left (c x^n\right )-195 b^3 e^2 n^3 x^2+144 b^3 e^3 n x^3 \log ^2\left (c x^n\right )-270 b^3 e^2 n x^2 \log ^2\left (c x^n\right )+864 b^3 e n x \log ^2\left (c x^n\right )+1920 b^3 e n^3 x-144 b^3 e^3 n^2 x^3 \log \left (c x^n\right )+342 b^3 e^2 n^2 x^2 \log \left (c x^n\right )-1872 b^3 e n^2 x \log \left (c x^n\right )-48 b^3 n^3 \log (e x+1)-48 b^3 e^3 n^3 x^3 \log (e x+1)+216 b^3 \log ^3\left (c x^n\right ) \log (e x+1)+216 b^3 e^3 x^3 \log ^3\left (c x^n\right ) \log (e x+1)-216 b^3 e^3 n x^3 \log ^2\left (c x^n\right ) \log (e x+1)-216 b^3 n \log ^2\left (c x^n\right ) \log (e x+1)+144 b^3 e^3 n^2 x^3 \log \left (c x^n\right ) \log (e x+1)+144 b^3 n^2 \log \left (c x^n\right ) \log (e x+1)+72 b n \left (9 a^2-6 b n a+2 b^2 n^2+9 b^2 \log ^2\left (c x^n\right )-6 b (b n-3 a) \log \left (c x^n\right )\right ) \text {Li}_2(-e x)+432 b^2 n^2 \left (-3 a+b n-3 b \log \left (c x^n\right )\right ) \text {Li}_3(-e x)+1296 b^3 n^3 \text {Li}_4(-e x)}{648 e^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.80, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{3} x^{2} \log \left (c x^{n}\right )^{3} \log \left (e x + 1\right ) + 3 \, a b^{2} x^{2} \log \left (c x^{n}\right )^{2} \log \left (e x + 1\right ) + 3 \, a^{2} b x^{2} \log \left (c x^{n}\right ) \log \left (e x + 1\right ) + a^{3} x^{2} \log \left (e x + 1\right ), x\right ) \]
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 \[ \int {\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x^{2} \log \left (e x + 1\right )\,{d x} \]
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 \[ \int \left (b \ln \left (c \,x^{n}\right )+a \right )^{3} x^{2} \ln \left (e x +1\right )\, dx \]
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 \[ -\frac {{\left (2 \, b^{3} e^{3} x^{3} - 3 \, b^{3} e^{2} x^{2} + 6 \, b^{3} e x - 6 \, {\left (b^{3} e^{3} x^{3} + b^{3}\right )} \log \left (e x + 1\right )\right )} \log \left (x^{n}\right )^{3}}{18 \, e^{3}} + \frac {\frac {1}{3} \, {\left (6 \, x^{3} \log \left (e x + 1\right ) - e {\left (\frac {2 \, e^{2} x^{3} - 3 \, e x^{2} + 6 \, x}{e^{3}} - \frac {6 \, \log \left (e x + 1\right )}{e^{4}}\right )}\right )} b^{3} e^{3} \log \relax (c)^{3} + {\left (6 \, x^{3} \log \left (e x + 1\right ) - e {\left (\frac {2 \, e^{2} x^{3} - 3 \, e x^{2} + 6 \, x}{e^{3}} - \frac {6 \, \log \left (e x + 1\right )}{e^{4}}\right )}\right )} a b^{2} e^{3} \log \relax (c)^{2} + {\left (6 \, x^{3} \log \left (e x + 1\right ) - e {\left (\frac {2 \, e^{2} x^{3} - 3 \, e x^{2} + 6 \, x}{e^{3}} - \frac {6 \, \log \left (e x + 1\right )}{e^{4}}\right )}\right )} a^{2} b e^{3} \log \relax (c) + \frac {1}{3} \, {\left (6 \, x^{3} \log \left (e x + 1\right ) - e {\left (\frac {2 \, e^{2} x^{3} - 3 \, e x^{2} + 6 \, x}{e^{3}} - \frac {6 \, \log \left (e x + 1\right )}{e^{4}}\right )}\right )} a^{3} e^{3} + \int \frac {18 \, {\left (b^{3} e^{3} \log \relax (c)^{2} + 2 \, a b^{2} e^{3} \log \relax (c) + a^{2} b e^{3}\right )} x^{3} \log \left (e x + 1\right ) \log \left (x^{n}\right ) + {\left (2 \, b^{3} e^{3} n x^{3} - 3 \, b^{3} e^{2} n x^{2} + 6 \, b^{3} e n x - 6 \, {\left (b^{3} n - {\left (3 \, a b^{2} e^{3} - {\left (e^{3} n - 3 \, e^{3} \log \relax (c)\right )} b^{3}\right )} x^{3}\right )} \log \left (e x + 1\right )\right )} \log \left (x^{n}\right )^{2}}{x}\,{d x}}{6 \, e^{3}} \]
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 \[ \int x^2\,\ln \left (e\,x+1\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3 \,d x \]
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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