Optimal. Leaf size=710 \[ \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{21 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{64 e^2}+\frac{3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{32 e^3}-\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}{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{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{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{\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{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{45 b^3 n^3 x^2}{256 e^2}-\frac{255 b^3 n^3 x}{128 e^3}+\frac{3 b^3 n^3 \log (e x+1)}{128 e^4}-\frac{175 b^3 n^3 x^3}{3456 e}-\frac{3}{128} b^3 n^3 x^4 \log (e x+1)+\frac{3}{128} b^3 n^3 x^4 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.777342, antiderivative size = 710, normalized size of antiderivative = 1., 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 = {2395, 43, 2377, 2296, 2295, 2305, 2304, 2374, 2383, 6589, 2376, 2391} \[ \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{21 b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{64 e^2}+\frac{3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{32 e^3}-\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}{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{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{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{\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{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{45 b^3 n^3 x^2}{256 e^2}-\frac{255 b^3 n^3 x}{128 e^3}+\frac{3 b^3 n^3 \log (e x+1)}{128 e^4}-\frac{175 b^3 n^3 x^3}{3456 e}-\frac{3}{128} b^3 n^3 x^4 \log (e x+1)+\frac{3}{128} b^3 n^3 x^4 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2395
Rule 43
Rule 2377
Rule 2296
Rule 2295
Rule 2305
Rule 2304
Rule 2374
Rule 2383
Rule 6589
Rule 2376
Rule 2391
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*}
Mathematica [A] time = 0.338973, size = 1144, normalized size = 1.61 \[ \frac{-432 a^3 e^4 x^4+162 b^3 e^4 n^3 x^4-432 b^3 e^4 \log ^3\left (c x^n\right ) x^4-486 a b^2 e^4 n^2 x^4-1296 a b^2 e^4 \log ^2\left (c x^n\right ) x^4+648 b^3 e^4 n \log ^2\left (c x^n\right ) x^4+648 a^2 b e^4 n x^4-1296 a^2 b e^4 \log \left (c x^n\right ) x^4-486 b^3 e^4 n^2 \log \left (c x^n\right ) x^4+1296 a b^2 e^4 n \log \left (c x^n\right ) x^4+1728 a^3 e^4 \log (e x+1) x^4-162 b^3 e^4 n^3 \log (e x+1) x^4+1728 b^3 e^4 \log ^3\left (c x^n\right ) \log (e x+1) x^4+648 a b^2 e^4 n^2 \log (e x+1) x^4+5184 a b^2 e^4 \log ^2\left (c x^n\right ) \log (e x+1) x^4-1296 b^3 e^4 n \log ^2\left (c x^n\right ) \log (e x+1) x^4-1296 a^2 b e^4 n \log (e x+1) x^4+5184 a^2 b e^4 \log \left (c x^n\right ) \log (e x+1) x^4+648 b^3 e^4 n^2 \log \left (c x^n\right ) \log (e x+1) x^4-2592 a b^2 e^4 n \log \left (c x^n\right ) \log (e x+1) x^4+576 a^3 e^3 x^3-350 b^3 e^3 n^3 x^3+576 b^3 e^3 \log ^3\left (c x^n\right ) x^3+888 a b^2 e^3 n^2 x^3+1728 a b^2 e^3 \log ^2\left (c x^n\right ) x^3-1008 b^3 e^3 n \log ^2\left (c x^n\right ) x^3-1008 a^2 b e^3 n x^3+1728 a^2 b e^3 \log \left (c x^n\right ) x^3+888 b^3 e^3 n^2 \log \left (c x^n\right ) x^3-2016 a b^2 e^3 n \log \left (c x^n\right ) x^3+1215 b^3 e^2 n^3 x^2-864 b^3 e^2 \log ^3\left (c x^n\right ) x^2-864 a^3 e^2 x^2-2268 a b^2 e^2 n^2 x^2-2592 a b^2 e^2 \log ^2\left (c x^n\right ) x^2+1944 b^3 e^2 n \log ^2\left (c x^n\right ) x^2+1944 a^2 b e^2 n x^2-2592 a^2 b e^2 \log \left (c x^n\right ) x^2-2268 b^3 e^2 n^2 \log \left (c x^n\right ) x^2+3888 a b^2 e^2 n \log \left (c x^n\right ) x^2-13770 b^3 e n^3 x+1728 b^3 e \log ^3\left (c x^n\right ) x+13608 a b^2 e n^2 x+5184 a b^2 e \log ^2\left (c x^n\right ) x-6480 b^3 e n \log ^2\left (c x^n\right ) x+1728 a^3 e x-6480 a^2 b e n x+13608 b^3 e n^2 \log \left (c x^n\right ) x+5184 a^2 b e \log \left (c x^n\right ) x-12960 a b^2 e n \log \left (c x^n\right ) x-1728 a^3 \log (e x+1)+162 b^3 n^3 \log (e x+1)-1728 b^3 \log ^3\left (c x^n\right ) \log (e x+1)-648 a b^2 n^2 \log (e x+1)-5184 a b^2 \log ^2\left (c x^n\right ) \log (e x+1)+1296 b^3 n \log ^2\left (c x^n\right ) \log (e x+1)+1296 a^2 b n \log (e x+1)-648 b^3 n^2 \log \left (c x^n\right ) \log (e x+1)-5184 a^2 b \log \left (c x^n\right ) \log (e x+1)+2592 a b^2 n \log \left (c x^n\right ) \log (e x+1)-648 b n \left (8 a^2-4 b n a+b^2 n^2+8 b^2 \log ^2\left (c x^n\right )-4 b (b n-4 a) \log \left (c x^n\right )\right ) \text{PolyLog}(2,-e x)+2592 b^2 n^2 \left (4 a-b n+4 b \log \left (c x^n\right )\right ) \text{PolyLog}(3,-e x)-10368 b^3 n^3 \text{PolyLog}(4,-e x)}{6912 e^4} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.154, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{3}\ln \left ( ex+1 \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (3 \, b^{3} e^{4} x^{4} - 4 \, b^{3} e^{3} x^{3} + 6 \, b^{3} e^{2} x^{2} - 12 \, b^{3} e x - 12 \,{\left (b^{3} e^{4} x^{4} - b^{3}\right )} \log \left (e x + 1\right )\right )} \log \left (x^{n}\right )^{3}}{48 \, e^{4}} + \frac{\frac{1}{3} \,{\left (12 \, x^{4} \log \left (e x + 1\right ) - e{\left (\frac{3 \, e^{3} x^{4} - 4 \, e^{2} x^{3} + 6 \, e x^{2} - 12 \, x}{e^{4}} + \frac{12 \, \log \left (e x + 1\right )}{e^{5}}\right )}\right )} b^{3} e^{4} \log \left (c\right )^{3} +{\left (12 \, x^{4} \log \left (e x + 1\right ) - e{\left (\frac{3 \, e^{3} x^{4} - 4 \, e^{2} x^{3} + 6 \, e x^{2} - 12 \, x}{e^{4}} + \frac{12 \, \log \left (e x + 1\right )}{e^{5}}\right )}\right )} a b^{2} e^{4} \log \left (c\right )^{2} +{\left (12 \, x^{4} \log \left (e x + 1\right ) - e{\left (\frac{3 \, e^{3} x^{4} - 4 \, e^{2} x^{3} + 6 \, e x^{2} - 12 \, x}{e^{4}} + \frac{12 \, \log \left (e x + 1\right )}{e^{5}}\right )}\right )} a^{2} b e^{4} \log \left (c\right ) + \frac{1}{3} \,{\left (12 \, x^{4} \log \left (e x + 1\right ) - e{\left (\frac{3 \, e^{3} x^{4} - 4 \, e^{2} x^{3} + 6 \, e x^{2} - 12 \, x}{e^{4}} + \frac{12 \, \log \left (e x + 1\right )}{e^{5}}\right )}\right )} a^{3} e^{4} + \int \frac{48 \,{\left (b^{3} e^{4} \log \left (c\right )^{2} + 2 \, a b^{2} e^{4} \log \left (c\right ) + a^{2} b e^{4}\right )} x^{4} \log \left (e x + 1\right ) \log \left (x^{n}\right ) +{\left (3 \, b^{3} e^{4} n x^{4} - 4 \, b^{3} e^{3} n x^{3} + 6 \, b^{3} e^{2} n x^{2} - 12 \, b^{3} e n x + 12 \,{\left ({\left (4 \, a b^{2} e^{4} -{\left (e^{4} n - 4 \, e^{4} \log \left (c\right )\right )} b^{3}\right )} x^{4} + b^{3} n\right )} \log \left (e x + 1\right )\right )} \log \left (x^{n}\right )^{2}}{x}\,{d x}}{16 \, e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b^{3} x^{3} \log \left (c x^{n}\right )^{3} \log \left (e x + 1\right ) + 3 \, a b^{2} x^{3} \log \left (c x^{n}\right )^{2} \log \left (e x + 1\right ) + 3 \, a^{2} b x^{3} \log \left (c x^{n}\right ) \log \left (e x + 1\right ) + a^{3} x^{3} \log \left (e x + 1\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x^{3} \log \left (e x + 1\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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