Optimal. Leaf size=456 \[ -\frac {b n \text {Li}_2(-e x) \left (a+b \log \left (c x^n\right )\right )}{2 e^4}-\frac {\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{4 e^4}+\frac {b n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{8 e^4}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{4 e^3}-\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{8 e^3}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}+\frac {3 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{16 e^2}+\frac {1}{4} x^4 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{8} b n x^4 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}-\frac {7 b n x^3 \left (a+b \log \left (c x^n\right )\right )}{72 e}-\frac {1}{16} x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {a b n x}{2 e^3}-\frac {b^2 n x \log \left (c x^n\right )}{2 e^3}+\frac {b^2 n^2 \text {Li}_2(-e x)}{8 e^4}+\frac {b^2 n^2 \text {Li}_3(-e x)}{2 e^4}-\frac {b^2 n^2 \log (e x+1)}{32 e^4}+\frac {21 b^2 n^2 x}{32 e^3}-\frac {7 b^2 n^2 x^2}{64 e^2}+\frac {1}{32} b^2 n^2 x^4 \log (e x+1)+\frac {37 b^2 n^2 x^3}{864 e}-\frac {3}{128} b^2 n^2 x^4 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.33, antiderivative size = 456, normalized size of antiderivative = 1.00, number of steps used = 15, 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} \[ -\frac {b n \text {PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )}{2 e^4}+\frac {b^2 n^2 \text {PolyLog}(2,-e x)}{8 e^4}+\frac {b^2 n^2 \text {PolyLog}(3,-e x)}{2 e^4}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}+\frac {3 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{16 e^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{4 e^3}-\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{8 e^3}-\frac {\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{4 e^4}+\frac {b n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{8 e^4}+\frac {1}{4} x^4 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{8} b n x^4 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}-\frac {7 b n x^3 \left (a+b \log \left (c x^n\right )\right )}{72 e}-\frac {1}{16} x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {a b n x}{2 e^3}-\frac {b^2 n x \log \left (c x^n\right )}{2 e^3}-\frac {7 b^2 n^2 x^2}{64 e^2}+\frac {21 b^2 n^2 x}{32 e^3}-\frac {b^2 n^2 \log (e x+1)}{32 e^4}+\frac {37 b^2 n^2 x^3}{864 e}+\frac {1}{32} b^2 n^2 x^4 \log (e x+1)-\frac {3}{128} b^2 n^2 x^4 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 2295
Rule 2304
Rule 2374
Rule 2376
Rule 2377
Rule 2391
Rule 2395
Rule 6589
Rubi steps
\begin {align*} \int x^3 \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}{4 e^3}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}-\frac {1}{16} x^4 \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}+\frac {1}{4} x^4 \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 )}{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 {x \left (a+b \log \left (c x^n\right )\right )^2}{4 e^3}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}-\frac {1}{16} x^4 \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}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)+\frac {1}{8} (b n) \int x^3 \left (a+b \log \left (c x^n\right )\right ) \, dx-\frac {1}{2} (b n) \int x^3 \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}{2 e^4}-\frac {(b n) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{2 e^3}+\frac {(b n) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{4 e^2}-\frac {(b n) \int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx}{6 e}\\ &=-\frac {a b n x}{2 e^3}-\frac {b^2 n^2 x^2}{16 e^2}+\frac {b^2 n^2 x^3}{54 e}-\frac {1}{128} b^2 n^2 x^4-\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{8 e^3}+\frac {3 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{16 e^2}-\frac {7 b n x^3 \left (a+b \log \left (c x^n\right )\right )}{72 e}+\frac {1}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{4 e^3}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}-\frac {1}{16} x^4 \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)}{8 e^4}-\frac {1}{8} b n x^4 \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)}{4 e^4}+\frac {1}{4} x^4 \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)}{2 e^4}-\frac {\left (b^2 n\right ) \int \log \left (c x^n\right ) \, dx}{2 e^3}+\frac {1}{2} \left (b^2 n^2\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 (b^2 n^2\right ) \int \frac {\text {Li}_2(-e x)}{x} \, dx}{2 e^4}\\ &=-\frac {a b n x}{2 e^3}+\frac {5 b^2 n^2 x}{8 e^3}-\frac {3 b^2 n^2 x^2}{32 e^2}+\frac {7 b^2 n^2 x^3}{216 e}-\frac {1}{64} b^2 n^2 x^4-\frac {b^2 n x \log \left (c x^n\right )}{2 e^3}-\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{8 e^3}+\frac {3 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{16 e^2}-\frac {7 b n x^3 \left (a+b \log \left (c x^n\right )\right )}{72 e}+\frac {1}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{4 e^3}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}-\frac {1}{16} x^4 \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)}{8 e^4}-\frac {1}{8} b n x^4 \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)}{4 e^4}+\frac {1}{4} x^4 \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)}{2 e^4}+\frac {b^2 n^2 \text {Li}_3(-e x)}{2 e^4}+\frac {1}{8} \left (b^2 n^2\right ) \int x^3 \log (1+e x) \, dx-\frac {\left (b^2 n^2\right ) \int \frac {\log (1+e x)}{x} \, dx}{8 e^4}\\ &=-\frac {a b n x}{2 e^3}+\frac {5 b^2 n^2 x}{8 e^3}-\frac {3 b^2 n^2 x^2}{32 e^2}+\frac {7 b^2 n^2 x^3}{216 e}-\frac {1}{64} b^2 n^2 x^4-\frac {b^2 n x \log \left (c x^n\right )}{2 e^3}-\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{8 e^3}+\frac {3 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{16 e^2}-\frac {7 b n x^3 \left (a+b \log \left (c x^n\right )\right )}{72 e}+\frac {1}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{4 e^3}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}-\frac {1}{16} x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{32} b^2 n^2 x^4 \log (1+e x)+\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{8 e^4}-\frac {1}{8} b n x^4 \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)}{4 e^4}+\frac {1}{4} x^4 \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)}{8 e^4}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{2 e^4}+\frac {b^2 n^2 \text {Li}_3(-e x)}{2 e^4}-\frac {1}{32} \left (b^2 e n^2\right ) \int \frac {x^4}{1+e x} \, dx\\ &=-\frac {a b n x}{2 e^3}+\frac {5 b^2 n^2 x}{8 e^3}-\frac {3 b^2 n^2 x^2}{32 e^2}+\frac {7 b^2 n^2 x^3}{216 e}-\frac {1}{64} b^2 n^2 x^4-\frac {b^2 n x \log \left (c x^n\right )}{2 e^3}-\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{8 e^3}+\frac {3 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{16 e^2}-\frac {7 b n x^3 \left (a+b \log \left (c x^n\right )\right )}{72 e}+\frac {1}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{4 e^3}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}-\frac {1}{16} x^4 \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{32} b^2 n^2 x^4 \log (1+e x)+\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{8 e^4}-\frac {1}{8} b n x^4 \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)}{4 e^4}+\frac {1}{4} x^4 \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)}{8 e^4}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{2 e^4}+\frac {b^2 n^2 \text {Li}_3(-e x)}{2 e^4}-\frac {1}{32} \left (b^2 e n^2\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 {a b n x}{2 e^3}+\frac {21 b^2 n^2 x}{32 e^3}-\frac {7 b^2 n^2 x^2}{64 e^2}+\frac {37 b^2 n^2 x^3}{864 e}-\frac {3}{128} b^2 n^2 x^4-\frac {b^2 n x \log \left (c x^n\right )}{2 e^3}-\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{8 e^3}+\frac {3 b n x^2 \left (a+b \log \left (c x^n\right )\right )}{16 e^2}-\frac {7 b n x^3 \left (a+b \log \left (c x^n\right )\right )}{72 e}+\frac {1}{16} b n x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{4 e^3}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2}+\frac {x^3 \left (a+b \log \left (c x^n\right )\right )^2}{12 e}-\frac {1}{16} x^4 \left (a+b \log \left (c x^n\right )\right )^2-\frac {b^2 n^2 \log (1+e x)}{32 e^4}+\frac {1}{32} b^2 n^2 x^4 \log (1+e x)+\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{8 e^4}-\frac {1}{8} b n x^4 \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)}{4 e^4}+\frac {1}{4} x^4 \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)}{8 e^4}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{2 e^4}+\frac {b^2 n^2 \text {Li}_3(-e x)}{2 e^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.21, size = 594, normalized size = 1.30 \[ \frac {-216 a^2 e^4 x^4+864 a^2 e^4 x^4 \log (e x+1)+288 a^2 e^3 x^3-432 a^2 e^2 x^2+864 a^2 e x-864 a^2 \log (e x+1)-432 a b e^4 x^4 \log \left (c x^n\right )+1728 a b e^4 x^4 \log (e x+1) \log \left (c x^n\right )+576 a b e^3 x^3 \log \left (c x^n\right )-864 a b e^2 x^2 \log \left (c x^n\right )+432 b n \text {Li}_2(-e x) \left (-4 a-4 b \log \left (c x^n\right )+b n\right )+1728 a b e x \log \left (c x^n\right )-1728 a b \log (e x+1) \log \left (c x^n\right )+216 a b e^4 n x^4-432 a b e^4 n x^4 \log (e x+1)-336 a b e^3 n x^3+648 a b e^2 n x^2-2160 a b e n x+432 a b n \log (e x+1)-216 b^2 e^4 x^4 \log ^2\left (c x^n\right )+864 b^2 e^4 x^4 \log (e x+1) \log ^2\left (c x^n\right )+216 b^2 e^4 n x^4 \log \left (c x^n\right )-432 b^2 e^4 n x^4 \log (e x+1) \log \left (c x^n\right )+288 b^2 e^3 x^3 \log ^2\left (c x^n\right )-336 b^2 e^3 n x^3 \log \left (c x^n\right )-432 b^2 e^2 x^2 \log ^2\left (c x^n\right )+648 b^2 e^2 n x^2 \log \left (c x^n\right )+864 b^2 e x \log ^2\left (c x^n\right )-864 b^2 \log (e x+1) \log ^2\left (c x^n\right )-2160 b^2 e n x \log \left (c x^n\right )+432 b^2 n \log (e x+1) \log \left (c x^n\right )-81 b^2 e^4 n^2 x^4+108 b^2 e^4 n^2 x^4 \log (e x+1)+148 b^2 e^3 n^2 x^3-378 b^2 e^2 n^2 x^2+1728 b^2 n^2 \text {Li}_3(-e x)+2268 b^2 e n^2 x-108 b^2 n^2 \log (e x+1)}{3456 e^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{2} x^{3} \log \left (c x^{n}\right )^{2} \log \left (e x + 1\right ) + 2 \, a b x^{3} \log \left (c x^{n}\right ) \log \left (e x + 1\right ) + a^{2} x^{3} \log \left (e x + 1\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x^{3} \log \left (e x + 1\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.57, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \,x^{n}\right )+a \right )^{2} x^{3} \ln \left (e x +1\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {{\left (3 \, b^{2} e^{4} x^{4} - 4 \, b^{2} e^{3} x^{3} + 6 \, b^{2} e^{2} x^{2} - 12 \, b^{2} e x - 12 \, {\left (b^{2} e^{4} x^{4} - b^{2}\right )} \log \left (e x + 1\right )\right )} \log \left (x^{n}\right )^{2}}{48 \, e^{4}} + \frac {-\frac {3}{16} \, b^{2} e^{4} n^{2} x^{4} + \frac {3}{4} \, b^{2} e^{4} n x^{4} \log \left (x^{n}\right ) + \frac {4}{9} \, b^{2} e^{3} n^{2} x^{3} - \frac {4}{3} \, b^{2} e^{3} n x^{3} \log \left (x^{n}\right ) + \frac {1}{2} \, {\left (12 \, x^{4} \log \left (e x + 1\right ) - e {\left (\frac {3 \, e^{3} x^{4} - 4 \, e^{2} x^{3} + 6 \, e x^{2} - 12 \, x}{e^{4}} + \frac {12 \, \log \left (e x + 1\right )}{e^{5}}\right )}\right )} b^{2} e^{4} \log \relax (c)^{2} - \frac {3}{2} \, b^{2} e^{2} n^{2} x^{2} + {\left (12 \, x^{4} \log \left (e x + 1\right ) - e {\left (\frac {3 \, e^{3} x^{4} - 4 \, e^{2} x^{3} + 6 \, e x^{2} - 12 \, x}{e^{4}} + \frac {12 \, \log \left (e x + 1\right )}{e^{5}}\right )}\right )} a b e^{4} \log \relax (c) + 3 \, b^{2} e^{2} n x^{2} \log \left (x^{n}\right ) + \frac {1}{2} \, {\left (12 \, x^{4} \log \left (e x + 1\right ) - e {\left (\frac {3 \, e^{3} x^{4} - 4 \, e^{2} x^{3} + 6 \, e x^{2} - 12 \, x}{e^{4}} + \frac {12 \, \log \left (e x + 1\right )}{e^{5}}\right )}\right )} a^{2} e^{4} + 12 \, b^{2} e n^{2} x - 12 \, b^{2} e n x \log \left (x^{n}\right ) + \int \frac {12 \, {\left ({\left (4 \, a b e^{4} - {\left (e^{4} n - 4 \, e^{4} \log \relax (c)\right )} b^{2}\right )} x^{4} + b^{2} n\right )} \log \left (e x + 1\right ) \log \left (x^{n}\right )}{x}\,{d x}}{24 \, e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^3\,\ln \left (e\,x+1\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________