Optimal. Leaf size=710 \[ \frac {15 a b^2 n^2 x}{8 e^3}-\frac {255 b^3 n^3 x}{128 e^3}+\frac {45 b^3 n^3 x^2}{256 e^2}-\frac {175 b^3 n^3 x^3}{3456 e}+\frac {3}{128} b^3 n^3 x^4+\frac {15 b^3 n^2 x \log \left (c x^n\right )}{8 e^3}+\frac {3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{32 e^3}-\frac {21 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{64 e^2}+\frac {37 b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right )}{288 e}-\frac {9}{128} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {15 b n x \left (a+b \log \left (c x^n\right )\right )^2}{16 e^3}+\frac {9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{32 e^2}-\frac {7 b n x^3 \left (a+b \log \left (c x^n\right )\right )^2}{48 e}+\frac {3}{32} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{4 e^3}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{8 e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^3}{12 e}-\frac {1}{16} x^4 \left (a+b \log \left (c x^n\right )\right )^3+\frac {3 b^3 n^3 \log (1+e x)}{128 e^4}-\frac {3}{128} b^3 n^3 x^4 \log (1+e x)-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{32 e^4}+\frac {3}{32} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{16 e^4}-\frac {3}{16} b n x^4 \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)}{4 e^4}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)-\frac {3 b^3 n^3 \text {Li}_2(-e x)}{32 e^4}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{8 e^4}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2(-e x)}{4 e^4}-\frac {3 b^3 n^3 \text {Li}_3(-e x)}{8 e^4}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(-e x)}{2 e^4}-\frac {3 b^3 n^3 \text {Li}_4(-e x)}{2 e^4} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.50, antiderivative size = 710, normalized size of antiderivative = 1.00, number of steps
used = 29, number of rules used = 12, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.546, Rules used = {2442, 45,
2424, 2333, 2332, 2342, 2341, 2421, 2430, 6724, 2423, 2438} \begin {gather*} \frac {3 b^2 n^2 \text {PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )}{8 e^4}+\frac {3 b^2 n^2 \text {PolyLog}(3,-e x) \left (a+b \log \left (c x^n\right )\right )}{2 e^4}-\frac {3 b n \text {PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )^2}{4 e^4}-\frac {3 b^3 n^3 \text {PolyLog}(2,-e x)}{32 e^4}-\frac {3 b^3 n^3 \text {PolyLog}(3,-e x)}{8 e^4}-\frac {3 b^3 n^3 \text {PolyLog}(4,-e x)}{2 e^4}-\frac {3 b^2 n^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{32 e^4}+\frac {3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{32 e^3}-\frac {21 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{64 e^2}+\frac {3}{32} b^2 n^2 x^4 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )+\frac {37 b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right )}{288 e}-\frac {9}{128} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {15 a b^2 n^2 x}{8 e^3}-\frac {\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3}{4 e^4}+\frac {3 b n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{16 e^4}+\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{4 e^3}-\frac {15 b n x \left (a+b \log \left (c x^n\right )\right )^2}{16 e^3}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{8 e^2}+\frac {9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{32 e^2}+\frac {1}{4} x^4 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^3-\frac {3}{16} b n x^4 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^3}{12 e}-\frac {7 b n x^3 \left (a+b \log \left (c x^n\right )\right )^2}{48 e}-\frac {1}{16} x^4 \left (a+b \log \left (c x^n\right )\right )^3+\frac {3}{32} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac {15 b^3 n^2 x \log \left (c x^n\right )}{8 e^3}+\frac {3 b^3 n^3 \log (e x+1)}{128 e^4}-\frac {255 b^3 n^3 x}{128 e^3}+\frac {45 b^3 n^3 x^2}{256 e^2}-\frac {3}{128} b^3 n^3 x^4 \log (e x+1)-\frac {175 b^3 n^3 x^3}{3456 e}+\frac {3}{128} b^3 n^3 x^4 \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 2332
Rule 2333
Rule 2341
Rule 2342
Rule 2421
Rule 2423
Rule 2424
Rule 2430
Rule 2438
Rule 2442
Rule 6724
Rubi steps
\begin {align*} \int x^3 \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}{4 e^3}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{8 e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^3}{12 e}-\frac {1}{16} x^4 \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)}{4 e^4}+\frac {1}{4} x^4 \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}{4 e^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}-\frac {1}{16} 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)}{4 e^4 x}+\frac {1}{4} x^3 \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}{4 e^3}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{8 e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^3}{12 e}-\frac {1}{16} x^4 \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)}{4 e^4}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)+\frac {1}{16} (3 b n) \int x^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx-\frac {1}{4} (3 b n) \int x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x) \, dx+\frac {(3 b n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x} \, dx}{4 e^4}-\frac {(3 b n) \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{4 e^3}+\frac {(3 b n) \int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{8 e^2}-\frac {(b n) \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{4 e}\\ &=-\frac {15 b n x \left (a+b \log \left (c x^n\right )\right )^2}{16 e^3}+\frac {9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{32 e^2}-\frac {7 b n x^3 \left (a+b \log \left (c x^n\right )\right )^2}{48 e}+\frac {3}{32} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{4 e^3}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{8 e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^3}{12 e}-\frac {1}{16} x^4 \left (a+b \log \left (c x^n\right )\right )^3+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{16 e^4}-\frac {3}{16} b n x^4 \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)}{4 e^4}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2(-e x)}{4 e^4}-\frac {1}{32} \left (3 b^2 n^2\right ) \int x^3 \left (a+b \log \left (c x^n\right )\right ) \, dx+\frac {1}{2} \left (3 b^2 n^2\right ) \int \left (\frac {a+b \log \left (c x^n\right )}{4 e^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{8 e^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{12 e}-\frac {1}{16} 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)}{4 e^4 x}+\frac {1}{4} x^3 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)\right ) \, dx+\frac {\left (3 b^2 n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{x} \, dx}{2 e^4}+\frac {\left (3 b^2 n^2\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{2 e^3}-\frac {\left (3 b^2 n^2\right ) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{8 e^2}+\frac {\left (b^2 n^2\right ) \int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx}{6 e}\\ &=\frac {3 a b^2 n^2 x}{2 e^3}+\frac {3 b^3 n^3 x^2}{32 e^2}-\frac {b^3 n^3 x^3}{54 e}+\frac {3}{512} b^3 n^3 x^4-\frac {3 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{16 e^2}+\frac {b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right )}{18 e}-\frac {3}{128} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {15 b n x \left (a+b \log \left (c x^n\right )\right )^2}{16 e^3}+\frac {9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{32 e^2}-\frac {7 b n x^3 \left (a+b \log \left (c x^n\right )\right )^2}{48 e}+\frac {3}{32} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{4 e^3}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{8 e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^3}{12 e}-\frac {1}{16} x^4 \left (a+b \log \left (c x^n\right )\right )^3+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{16 e^4}-\frac {3}{16} b n x^4 \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)}{4 e^4}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2(-e x)}{4 e^4}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(-e x)}{2 e^4}-\frac {1}{32} \left (3 b^2 n^2\right ) \int x^3 \left (a+b \log \left (c x^n\right )\right ) \, dx+\frac {1}{8} \left (3 b^2 n^2\right ) \int x^3 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x) \, dx-\frac {\left (3 b^2 n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{x} \, dx}{8 e^4}+\frac {\left (3 b^2 n^2\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{8 e^3}+\frac {\left (3 b^3 n^2\right ) \int \log \left (c x^n\right ) \, dx}{2 e^3}-\frac {\left (3 b^2 n^2\right ) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{16 e^2}+\frac {\left (b^2 n^2\right ) \int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx}{8 e}-\frac {\left (3 b^3 n^3\right ) \int \frac {\text {Li}_3(-e x)}{x} \, dx}{2 e^4}\\ &=\frac {15 a b^2 n^2 x}{8 e^3}-\frac {3 b^3 n^3 x}{2 e^3}+\frac {9 b^3 n^3 x^2}{64 e^2}-\frac {7 b^3 n^3 x^3}{216 e}+\frac {3}{256} b^3 n^3 x^4+\frac {3 b^3 n^2 x \log \left (c x^n\right )}{2 e^3}+\frac {3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{32 e^3}-\frac {21 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{64 e^2}+\frac {37 b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right )}{288 e}-\frac {9}{128} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {15 b n x \left (a+b \log \left (c x^n\right )\right )^2}{16 e^3}+\frac {9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{32 e^2}-\frac {7 b n x^3 \left (a+b \log \left (c x^n\right )\right )^2}{48 e}+\frac {3}{32} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{4 e^3}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{8 e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^3}{12 e}-\frac {1}{16} x^4 \left (a+b \log \left (c x^n\right )\right )^3-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{32 e^4}+\frac {3}{32} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{16 e^4}-\frac {3}{16} b n x^4 \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)}{4 e^4}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{8 e^4}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2(-e x)}{4 e^4}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(-e x)}{2 e^4}-\frac {3 b^3 n^3 \text {Li}_4(-e x)}{2 e^4}+\frac {\left (3 b^3 n^2\right ) \int \log \left (c x^n\right ) \, dx}{8 e^3}-\frac {1}{8} \left (3 b^3 n^3\right ) \int \left (\frac {1}{4 e^3}-\frac {x}{8 e^2}+\frac {x^2}{12 e}-\frac {x^3}{16}-\frac {\log (1+e x)}{4 e^4 x}+\frac {1}{4} x^3 \log (1+e x)\right ) \, dx-\frac {\left (3 b^3 n^3\right ) \int \frac {\text {Li}_2(-e x)}{x} \, dx}{8 e^4}\\ &=\frac {15 a b^2 n^2 x}{8 e^3}-\frac {63 b^3 n^3 x}{32 e^3}+\frac {21 b^3 n^3 x^2}{128 e^2}-\frac {37 b^3 n^3 x^3}{864 e}+\frac {9}{512} b^3 n^3 x^4+\frac {15 b^3 n^2 x \log \left (c x^n\right )}{8 e^3}+\frac {3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{32 e^3}-\frac {21 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{64 e^2}+\frac {37 b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right )}{288 e}-\frac {9}{128} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {15 b n x \left (a+b \log \left (c x^n\right )\right )^2}{16 e^3}+\frac {9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{32 e^2}-\frac {7 b n x^3 \left (a+b \log \left (c x^n\right )\right )^2}{48 e}+\frac {3}{32} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{4 e^3}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{8 e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^3}{12 e}-\frac {1}{16} x^4 \left (a+b \log \left (c x^n\right )\right )^3-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{32 e^4}+\frac {3}{32} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{16 e^4}-\frac {3}{16} b n x^4 \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)}{4 e^4}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{8 e^4}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2(-e x)}{4 e^4}-\frac {3 b^3 n^3 \text {Li}_3(-e x)}{8 e^4}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(-e x)}{2 e^4}-\frac {3 b^3 n^3 \text {Li}_4(-e x)}{2 e^4}-\frac {1}{32} \left (3 b^3 n^3\right ) \int x^3 \log (1+e x) \, dx+\frac {\left (3 b^3 n^3\right ) \int \frac {\log (1+e x)}{x} \, dx}{32 e^4}\\ &=\frac {15 a b^2 n^2 x}{8 e^3}-\frac {63 b^3 n^3 x}{32 e^3}+\frac {21 b^3 n^3 x^2}{128 e^2}-\frac {37 b^3 n^3 x^3}{864 e}+\frac {9}{512} b^3 n^3 x^4+\frac {15 b^3 n^2 x \log \left (c x^n\right )}{8 e^3}+\frac {3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{32 e^3}-\frac {21 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{64 e^2}+\frac {37 b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right )}{288 e}-\frac {9}{128} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {15 b n x \left (a+b \log \left (c x^n\right )\right )^2}{16 e^3}+\frac {9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{32 e^2}-\frac {7 b n x^3 \left (a+b \log \left (c x^n\right )\right )^2}{48 e}+\frac {3}{32} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{4 e^3}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{8 e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^3}{12 e}-\frac {1}{16} x^4 \left (a+b \log \left (c x^n\right )\right )^3-\frac {3}{128} b^3 n^3 x^4 \log (1+e x)-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{32 e^4}+\frac {3}{32} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{16 e^4}-\frac {3}{16} b n x^4 \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)}{4 e^4}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)-\frac {3 b^3 n^3 \text {Li}_2(-e x)}{32 e^4}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{8 e^4}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2(-e x)}{4 e^4}-\frac {3 b^3 n^3 \text {Li}_3(-e x)}{8 e^4}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(-e x)}{2 e^4}-\frac {3 b^3 n^3 \text {Li}_4(-e x)}{2 e^4}+\frac {1}{128} \left (3 b^3 e n^3\right ) \int \frac {x^4}{1+e x} \, dx\\ &=\frac {15 a b^2 n^2 x}{8 e^3}-\frac {63 b^3 n^3 x}{32 e^3}+\frac {21 b^3 n^3 x^2}{128 e^2}-\frac {37 b^3 n^3 x^3}{864 e}+\frac {9}{512} b^3 n^3 x^4+\frac {15 b^3 n^2 x \log \left (c x^n\right )}{8 e^3}+\frac {3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{32 e^3}-\frac {21 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{64 e^2}+\frac {37 b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right )}{288 e}-\frac {9}{128} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {15 b n x \left (a+b \log \left (c x^n\right )\right )^2}{16 e^3}+\frac {9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{32 e^2}-\frac {7 b n x^3 \left (a+b \log \left (c x^n\right )\right )^2}{48 e}+\frac {3}{32} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{4 e^3}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{8 e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^3}{12 e}-\frac {1}{16} x^4 \left (a+b \log \left (c x^n\right )\right )^3-\frac {3}{128} b^3 n^3 x^4 \log (1+e x)-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{32 e^4}+\frac {3}{32} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{16 e^4}-\frac {3}{16} b n x^4 \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)}{4 e^4}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)-\frac {3 b^3 n^3 \text {Li}_2(-e x)}{32 e^4}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{8 e^4}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2(-e x)}{4 e^4}-\frac {3 b^3 n^3 \text {Li}_3(-e x)}{8 e^4}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(-e x)}{2 e^4}-\frac {3 b^3 n^3 \text {Li}_4(-e x)}{2 e^4}+\frac {1}{128} \left (3 b^3 e n^3\right ) \int \left (-\frac {1}{e^4}+\frac {x}{e^3}-\frac {x^2}{e^2}+\frac {x^3}{e}+\frac {1}{e^4 (1+e x)}\right ) \, dx\\ &=\frac {15 a b^2 n^2 x}{8 e^3}-\frac {255 b^3 n^3 x}{128 e^3}+\frac {45 b^3 n^3 x^2}{256 e^2}-\frac {175 b^3 n^3 x^3}{3456 e}+\frac {3}{128} b^3 n^3 x^4+\frac {15 b^3 n^2 x \log \left (c x^n\right )}{8 e^3}+\frac {3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{32 e^3}-\frac {21 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{64 e^2}+\frac {37 b^2 n^2 x^3 \left (a+b \log \left (c x^n\right )\right )}{288 e}-\frac {9}{128} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {15 b n x \left (a+b \log \left (c x^n\right )\right )^2}{16 e^3}+\frac {9 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{32 e^2}-\frac {7 b n x^3 \left (a+b \log \left (c x^n\right )\right )^2}{48 e}+\frac {3}{32} b n x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{4 e^3}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{8 e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^3}{12 e}-\frac {1}{16} x^4 \left (a+b \log \left (c x^n\right )\right )^3+\frac {3 b^3 n^3 \log (1+e x)}{128 e^4}-\frac {3}{128} b^3 n^3 x^4 \log (1+e x)-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{32 e^4}+\frac {3}{32} b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{16 e^4}-\frac {3}{16} b n x^4 \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)}{4 e^4}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)-\frac {3 b^3 n^3 \text {Li}_2(-e x)}{32 e^4}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{8 e^4}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2(-e x)}{4 e^4}-\frac {3 b^3 n^3 \text {Li}_3(-e x)}{8 e^4}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(-e x)}{2 e^4}-\frac {3 b^3 n^3 \text {Li}_4(-e x)}{2 e^4}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 1144, normalized size = 1.61 \begin {gather*} \frac {1728 a^3 e x-6480 a^2 b e n x+13608 a b^2 e n^2 x-13770 b^3 e n^3 x-864 a^3 e^2 x^2+1944 a^2 b e^2 n x^2-2268 a b^2 e^2 n^2 x^2+1215 b^3 e^2 n^3 x^2+576 a^3 e^3 x^3-1008 a^2 b e^3 n x^3+888 a b^2 e^3 n^2 x^3-350 b^3 e^3 n^3 x^3-432 a^3 e^4 x^4+648 a^2 b e^4 n x^4-486 a b^2 e^4 n^2 x^4+162 b^3 e^4 n^3 x^4+5184 a^2 b e x \log \left (c x^n\right )-12960 a b^2 e n x \log \left (c x^n\right )+13608 b^3 e n^2 x \log \left (c x^n\right )-2592 a^2 b e^2 x^2 \log \left (c x^n\right )+3888 a b^2 e^2 n x^2 \log \left (c x^n\right )-2268 b^3 e^2 n^2 x^2 \log \left (c x^n\right )+1728 a^2 b e^3 x^3 \log \left (c x^n\right )-2016 a b^2 e^3 n x^3 \log \left (c x^n\right )+888 b^3 e^3 n^2 x^3 \log \left (c x^n\right )-1296 a^2 b e^4 x^4 \log \left (c x^n\right )+1296 a b^2 e^4 n x^4 \log \left (c x^n\right )-486 b^3 e^4 n^2 x^4 \log \left (c x^n\right )+5184 a b^2 e x \log ^2\left (c x^n\right )-6480 b^3 e n x \log ^2\left (c x^n\right )-2592 a b^2 e^2 x^2 \log ^2\left (c x^n\right )+1944 b^3 e^2 n x^2 \log ^2\left (c x^n\right )+1728 a b^2 e^3 x^3 \log ^2\left (c x^n\right )-1008 b^3 e^3 n x^3 \log ^2\left (c x^n\right )-1296 a b^2 e^4 x^4 \log ^2\left (c x^n\right )+648 b^3 e^4 n x^4 \log ^2\left (c x^n\right )+1728 b^3 e x \log ^3\left (c x^n\right )-864 b^3 e^2 x^2 \log ^3\left (c x^n\right )+576 b^3 e^3 x^3 \log ^3\left (c x^n\right )-432 b^3 e^4 x^4 \log ^3\left (c x^n\right )-1728 a^3 \log (1+e x)+1296 a^2 b n \log (1+e x)-648 a b^2 n^2 \log (1+e x)+162 b^3 n^3 \log (1+e x)+1728 a^3 e^4 x^4 \log (1+e x)-1296 a^2 b e^4 n x^4 \log (1+e x)+648 a b^2 e^4 n^2 x^4 \log (1+e x)-162 b^3 e^4 n^3 x^4 \log (1+e x)-5184 a^2 b \log \left (c x^n\right ) \log (1+e x)+2592 a b^2 n \log \left (c x^n\right ) \log (1+e x)-648 b^3 n^2 \log \left (c x^n\right ) \log (1+e x)+5184 a^2 b e^4 x^4 \log \left (c x^n\right ) \log (1+e x)-2592 a b^2 e^4 n x^4 \log \left (c x^n\right ) \log (1+e x)+648 b^3 e^4 n^2 x^4 \log \left (c x^n\right ) \log (1+e x)-5184 a b^2 \log ^2\left (c x^n\right ) \log (1+e x)+1296 b^3 n \log ^2\left (c x^n\right ) \log (1+e x)+5184 a b^2 e^4 x^4 \log ^2\left (c x^n\right ) \log (1+e x)-1296 b^3 e^4 n x^4 \log ^2\left (c x^n\right ) \log (1+e x)-1728 b^3 \log ^3\left (c x^n\right ) \log (1+e x)+1728 b^3 e^4 x^4 \log ^3\left (c x^n\right ) \log (1+e x)-648 b n \left (8 a^2-4 a b n+b^2 n^2-4 b (-4 a+b n) \log \left (c x^n\right )+8 b^2 \log ^2\left (c x^n\right )\right ) \text {Li}_2(-e x)+2592 b^2 n^2 \left (4 a-b n+4 b \log \left (c x^n\right )\right ) \text {Li}_3(-e x)-10368 b^3 n^3 \text {Li}_4(-e x)}{6912 e^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int x^{3} \left (a +b \ln \left (c \,x^{n}\right )\right )^{3} \ln \left (e x +1\right )\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x^3\,\ln \left (e\,x+1\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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