Optimal. Leaf size=977 \[ \text{result too large to display} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.49895, antiderivative size = 977, normalized size of antiderivative = 1., number of steps used = 42, number of rules used = 17, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.68, Rules used = {2296, 2295, 2371, 6, 321, 205, 2351, 2324, 12, 4848, 2391, 2353, 2330, 2317, 2374, 6589, 2383} \[ 36 m n^3 x b^3-36 m n^2 x \log \left (c x^n\right ) b^3+\frac{12 \sqrt{e} m n^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \log \left (c x^n\right ) b^3}{\sqrt{f}}-6 n^3 x \log \left (d \left (f x^2+e\right )^m\right ) b^3+6 n^2 x \log \left (c x^n\right ) \log \left (d \left (f x^2+e\right )^m\right ) b^3-\frac{6 i \sqrt{e} m n^3 \text{PolyLog}\left (2,-\frac{i \sqrt{f} x}{\sqrt{e}}\right ) b^3}{\sqrt{f}}+\frac{6 i \sqrt{e} m n^3 \text{PolyLog}\left (2,\frac{i \sqrt{f} x}{\sqrt{e}}\right ) b^3}{\sqrt{f}}+\frac{6 \sqrt{-e} m n^3 \text{PolyLog}\left (3,-\frac{\sqrt{f} x}{\sqrt{-e}}\right ) b^3}{\sqrt{f}}-\frac{6 \sqrt{-e} m n^3 \text{PolyLog}\left (3,\frac{\sqrt{f} x}{\sqrt{-e}}\right ) b^3}{\sqrt{f}}+\frac{6 \sqrt{-e} m n^3 \text{PolyLog}\left (4,-\frac{\sqrt{f} x}{\sqrt{-e}}\right ) b^3}{\sqrt{f}}-\frac{6 \sqrt{-e} m n^3 \text{PolyLog}\left (4,\frac{\sqrt{f} x}{\sqrt{-e}}\right ) b^3}{\sqrt{f}}-24 a m n^2 x b^2-12 m n^2 (a-b n) x b^2+\frac{12 \sqrt{e} m n^2 (a-b n) \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) b^2}{\sqrt{f}}+6 a n^2 x \log \left (d \left (f x^2+e\right )^m\right ) b^2-\frac{6 \sqrt{-e} m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{PolyLog}\left (2,-\frac{\sqrt{f} x}{\sqrt{-e}}\right ) b^2}{\sqrt{f}}+\frac{6 \sqrt{-e} m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{PolyLog}\left (2,\frac{\sqrt{f} x}{\sqrt{-e}}\right ) b^2}{\sqrt{f}}-\frac{6 \sqrt{-e} m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{PolyLog}\left (3,-\frac{\sqrt{f} x}{\sqrt{-e}}\right ) b^2}{\sqrt{f}}+\frac{6 \sqrt{-e} m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{PolyLog}\left (3,\frac{\sqrt{f} x}{\sqrt{-e}}\right ) b^2}{\sqrt{f}}+12 m n x \left (a+b \log \left (c x^n\right )\right )^2 b+\frac{3 \sqrt{-e} m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac{\sqrt{f} x}{\sqrt{-e}}\right ) b}{\sqrt{f}}-\frac{3 \sqrt{-e} m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (\frac{\sqrt{f} x}{\sqrt{-e}}+1\right ) b}{\sqrt{f}}-3 n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (f x^2+e\right )^m\right ) b+\frac{3 \sqrt{-e} m n \left (a+b \log \left (c x^n\right )\right )^2 \text{PolyLog}\left (2,-\frac{\sqrt{f} x}{\sqrt{-e}}\right ) b}{\sqrt{f}}-\frac{3 \sqrt{-e} m n \left (a+b \log \left (c x^n\right )\right )^2 \text{PolyLog}\left (2,\frac{\sqrt{f} x}{\sqrt{-e}}\right ) b}{\sqrt{f}}-2 m x \left (a+b \log \left (c x^n\right )\right )^3-\frac{\sqrt{-e} m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{f}}+\frac{\sqrt{-e} m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (\frac{\sqrt{f} x}{\sqrt{-e}}+1\right )}{\sqrt{f}}+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (f x^2+e\right )^m\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2296
Rule 2295
Rule 2371
Rule 6
Rule 321
Rule 205
Rule 2351
Rule 2324
Rule 12
Rule 4848
Rule 2391
Rule 2353
Rule 2330
Rule 2317
Rule 2374
Rule 6589
Rule 2383
Rubi steps
\begin{align*} \int \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right ) \, dx &=6 a b^2 n^2 x \log \left (d \left (e+f x^2\right )^m\right )-6 b^3 n^3 x \log \left (d \left (e+f x^2\right )^m\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )-(2 f m) \int \left (\frac{6 a b^2 n^2 x^2}{e+f x^2}-\frac{6 b^3 n^3 x^2}{e+f x^2}+\frac{6 b^3 n^2 x^2 \log \left (c x^n\right )}{e+f x^2}-\frac{3 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{e+f x^2}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^3}{e+f x^2}\right ) \, dx\\ &=6 a b^2 n^2 x \log \left (d \left (e+f x^2\right )^m\right )-6 b^3 n^3 x \log \left (d \left (e+f x^2\right )^m\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )-(2 f m) \int \left (\frac{\left (6 a b^2 n^2-6 b^3 n^3\right ) x^2}{e+f x^2}+\frac{6 b^3 n^2 x^2 \log \left (c x^n\right )}{e+f x^2}-\frac{3 b n x^2 \left (a+b \log \left (c x^n\right )\right )^2}{e+f x^2}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )^3}{e+f x^2}\right ) \, dx\\ &=6 a b^2 n^2 x \log \left (d \left (e+f x^2\right )^m\right )-6 b^3 n^3 x \log \left (d \left (e+f x^2\right )^m\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )-(2 f m) \int \frac{x^2 \left (a+b \log \left (c x^n\right )\right )^3}{e+f x^2} \, dx+(6 b f m n) \int \frac{x^2 \left (a+b \log \left (c x^n\right )\right )^2}{e+f x^2} \, dx-\left (12 b^3 f m n^2\right ) \int \frac{x^2 \log \left (c x^n\right )}{e+f x^2} \, dx-\left (12 b^2 f m n^2 (a-b n)\right ) \int \frac{x^2}{e+f x^2} \, dx\\ &=-12 b^2 m n^2 (a-b n) x+6 a b^2 n^2 x \log \left (d \left (e+f x^2\right )^m\right )-6 b^3 n^3 x \log \left (d \left (e+f x^2\right )^m\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )-(2 f m) \int \left (\frac{\left (a+b \log \left (c x^n\right )\right )^3}{f}-\frac{e \left (a+b \log \left (c x^n\right )\right )^3}{f \left (e+f x^2\right )}\right ) \, dx+(6 b f m n) \int \left (\frac{\left (a+b \log \left (c x^n\right )\right )^2}{f}-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{f \left (e+f x^2\right )}\right ) \, dx-\left (12 b^3 f m n^2\right ) \int \left (\frac{\log \left (c x^n\right )}{f}-\frac{e \log \left (c x^n\right )}{f \left (e+f x^2\right )}\right ) \, dx+\left (12 b^2 e m n^2 (a-b n)\right ) \int \frac{1}{e+f x^2} \, dx\\ &=-12 b^2 m n^2 (a-b n) x+\frac{12 b^2 \sqrt{e} m n^2 (a-b n) \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{f}}+6 a b^2 n^2 x \log \left (d \left (e+f x^2\right )^m\right )-6 b^3 n^3 x \log \left (d \left (e+f x^2\right )^m\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )-(2 m) \int \left (a+b \log \left (c x^n\right )\right )^3 \, dx+(2 e m) \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{e+f x^2} \, dx+(6 b m n) \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx-(6 b e m n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{e+f x^2} \, dx-\left (12 b^3 m n^2\right ) \int \log \left (c x^n\right ) \, dx+\left (12 b^3 e m n^2\right ) \int \frac{\log \left (c x^n\right )}{e+f x^2} \, dx\\ &=12 b^3 m n^3 x-12 b^2 m n^2 (a-b n) x+\frac{12 b^2 \sqrt{e} m n^2 (a-b n) \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{f}}-12 b^3 m n^2 x \log \left (c x^n\right )+\frac{12 b^3 \sqrt{e} m n^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \log \left (c x^n\right )}{\sqrt{f}}+6 b m n x \left (a+b \log \left (c x^n\right )\right )^2-2 m x \left (a+b \log \left (c x^n\right )\right )^3+6 a b^2 n^2 x \log \left (d \left (e+f x^2\right )^m\right )-6 b^3 n^3 x \log \left (d \left (e+f x^2\right )^m\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )+(2 e m) \int \left (\frac{\sqrt{-e} \left (a+b \log \left (c x^n\right )\right )^3}{2 e \left (\sqrt{-e}-\sqrt{f} x\right )}+\frac{\sqrt{-e} \left (a+b \log \left (c x^n\right )\right )^3}{2 e \left (\sqrt{-e}+\sqrt{f} x\right )}\right ) \, dx+(6 b m n) \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx-(6 b e m n) \int \left (\frac{\sqrt{-e} \left (a+b \log \left (c x^n\right )\right )^2}{2 e \left (\sqrt{-e}-\sqrt{f} x\right )}+\frac{\sqrt{-e} \left (a+b \log \left (c x^n\right )\right )^2}{2 e \left (\sqrt{-e}+\sqrt{f} x\right )}\right ) \, dx-\left (12 b^2 m n^2\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx-\left (12 b^3 e m n^3\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{e} \sqrt{f} x} \, dx\\ &=-12 a b^2 m n^2 x+12 b^3 m n^3 x-12 b^2 m n^2 (a-b n) x+\frac{12 b^2 \sqrt{e} m n^2 (a-b n) \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{f}}-12 b^3 m n^2 x \log \left (c x^n\right )+\frac{12 b^3 \sqrt{e} m n^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \log \left (c x^n\right )}{\sqrt{f}}+12 b m n x \left (a+b \log \left (c x^n\right )\right )^2-2 m x \left (a+b \log \left (c x^n\right )\right )^3+6 a b^2 n^2 x \log \left (d \left (e+f x^2\right )^m\right )-6 b^3 n^3 x \log \left (d \left (e+f x^2\right )^m\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )+\left (\sqrt{-e} m\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{\sqrt{-e}-\sqrt{f} x} \, dx+\left (\sqrt{-e} m\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{\sqrt{-e}+\sqrt{f} x} \, dx-\left (3 b \sqrt{-e} m n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\sqrt{-e}-\sqrt{f} x} \, dx-\left (3 b \sqrt{-e} m n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\sqrt{-e}+\sqrt{f} x} \, dx-\left (12 b^2 m n^2\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx-\left (12 b^3 m n^2\right ) \int \log \left (c x^n\right ) \, dx-\frac{\left (12 b^3 \sqrt{e} m n^3\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{x} \, dx}{\sqrt{f}}\\ &=-24 a b^2 m n^2 x+24 b^3 m n^3 x-12 b^2 m n^2 (a-b n) x+\frac{12 b^2 \sqrt{e} m n^2 (a-b n) \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{f}}-24 b^3 m n^2 x \log \left (c x^n\right )+\frac{12 b^3 \sqrt{e} m n^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \log \left (c x^n\right )}{\sqrt{f}}+12 b m n x \left (a+b \log \left (c x^n\right )\right )^2-2 m x \left (a+b \log \left (c x^n\right )\right )^3+\frac{3 b \sqrt{-e} m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{f}}-\frac{\sqrt{-e} m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{f}}-\frac{3 b \sqrt{-e} m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{f}}+\frac{\sqrt{-e} m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{f}}+6 a b^2 n^2 x \log \left (d \left (e+f x^2\right )^m\right )-6 b^3 n^3 x \log \left (d \left (e+f x^2\right )^m\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )+\frac{\left (3 b \sqrt{-e} m n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{x} \, dx}{\sqrt{f}}-\frac{\left (3 b \sqrt{-e} m n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{x} \, dx}{\sqrt{f}}-\left (12 b^3 m n^2\right ) \int \log \left (c x^n\right ) \, dx-\frac{\left (6 b^2 \sqrt{-e} m n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{x} \, dx}{\sqrt{f}}+\frac{\left (6 b^2 \sqrt{-e} m n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{x} \, dx}{\sqrt{f}}-\frac{\left (6 i b^3 \sqrt{e} m n^3\right ) \int \frac{\log \left (1-\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{x} \, dx}{\sqrt{f}}+\frac{\left (6 i b^3 \sqrt{e} m n^3\right ) \int \frac{\log \left (1+\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{x} \, dx}{\sqrt{f}}\\ &=-24 a b^2 m n^2 x+36 b^3 m n^3 x-12 b^2 m n^2 (a-b n) x+\frac{12 b^2 \sqrt{e} m n^2 (a-b n) \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{f}}-36 b^3 m n^2 x \log \left (c x^n\right )+\frac{12 b^3 \sqrt{e} m n^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \log \left (c x^n\right )}{\sqrt{f}}+12 b m n x \left (a+b \log \left (c x^n\right )\right )^2-2 m x \left (a+b \log \left (c x^n\right )\right )^3+\frac{3 b \sqrt{-e} m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{f}}-\frac{\sqrt{-e} m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{f}}-\frac{3 b \sqrt{-e} m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{f}}+\frac{\sqrt{-e} m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{f}}+6 a b^2 n^2 x \log \left (d \left (e+f x^2\right )^m\right )-6 b^3 n^3 x \log \left (d \left (e+f x^2\right )^m\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )-\frac{6 b^2 \sqrt{-e} m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{f}}+\frac{3 b \sqrt{-e} m n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{f}}+\frac{6 b^2 \sqrt{-e} m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{f}}-\frac{3 b \sqrt{-e} m n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{f}}-\frac{6 i b^3 \sqrt{e} m n^3 \text{Li}_2\left (-\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{f}}+\frac{6 i b^3 \sqrt{e} m n^3 \text{Li}_2\left (\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{f}}-\frac{\left (6 b^2 \sqrt{-e} m n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{x} \, dx}{\sqrt{f}}+\frac{\left (6 b^2 \sqrt{-e} m n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{x} \, dx}{\sqrt{f}}+\frac{\left (6 b^3 \sqrt{-e} m n^3\right ) \int \frac{\text{Li}_2\left (-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{x} \, dx}{\sqrt{f}}-\frac{\left (6 b^3 \sqrt{-e} m n^3\right ) \int \frac{\text{Li}_2\left (\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{x} \, dx}{\sqrt{f}}\\ &=-24 a b^2 m n^2 x+36 b^3 m n^3 x-12 b^2 m n^2 (a-b n) x+\frac{12 b^2 \sqrt{e} m n^2 (a-b n) \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{f}}-36 b^3 m n^2 x \log \left (c x^n\right )+\frac{12 b^3 \sqrt{e} m n^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \log \left (c x^n\right )}{\sqrt{f}}+12 b m n x \left (a+b \log \left (c x^n\right )\right )^2-2 m x \left (a+b \log \left (c x^n\right )\right )^3+\frac{3 b \sqrt{-e} m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{f}}-\frac{\sqrt{-e} m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{f}}-\frac{3 b \sqrt{-e} m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{f}}+\frac{\sqrt{-e} m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{f}}+6 a b^2 n^2 x \log \left (d \left (e+f x^2\right )^m\right )-6 b^3 n^3 x \log \left (d \left (e+f x^2\right )^m\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )-\frac{6 b^2 \sqrt{-e} m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{f}}+\frac{3 b \sqrt{-e} m n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{f}}+\frac{6 b^2 \sqrt{-e} m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{f}}-\frac{3 b \sqrt{-e} m n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{f}}-\frac{6 i b^3 \sqrt{e} m n^3 \text{Li}_2\left (-\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{f}}+\frac{6 i b^3 \sqrt{e} m n^3 \text{Li}_2\left (\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{f}}+\frac{6 b^3 \sqrt{-e} m n^3 \text{Li}_3\left (-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{f}}-\frac{6 b^2 \sqrt{-e} m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{f}}-\frac{6 b^3 \sqrt{-e} m n^3 \text{Li}_3\left (\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{f}}+\frac{6 b^2 \sqrt{-e} m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{f}}+\frac{\left (6 b^3 \sqrt{-e} m n^3\right ) \int \frac{\text{Li}_3\left (-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{x} \, dx}{\sqrt{f}}-\frac{\left (6 b^3 \sqrt{-e} m n^3\right ) \int \frac{\text{Li}_3\left (\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{x} \, dx}{\sqrt{f}}\\ &=-24 a b^2 m n^2 x+36 b^3 m n^3 x-12 b^2 m n^2 (a-b n) x+\frac{12 b^2 \sqrt{e} m n^2 (a-b n) \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{f}}-36 b^3 m n^2 x \log \left (c x^n\right )+\frac{12 b^3 \sqrt{e} m n^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \log \left (c x^n\right )}{\sqrt{f}}+12 b m n x \left (a+b \log \left (c x^n\right )\right )^2-2 m x \left (a+b \log \left (c x^n\right )\right )^3+\frac{3 b \sqrt{-e} m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{f}}-\frac{\sqrt{-e} m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{f}}-\frac{3 b \sqrt{-e} m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{f}}+\frac{\sqrt{-e} m \left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{f}}+6 a b^2 n^2 x \log \left (d \left (e+f x^2\right )^m\right )-6 b^3 n^3 x \log \left (d \left (e+f x^2\right )^m\right )+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-3 b n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )-\frac{6 b^2 \sqrt{-e} m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{f}}+\frac{3 b \sqrt{-e} m n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{f}}+\frac{6 b^2 \sqrt{-e} m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{f}}-\frac{3 b \sqrt{-e} m n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{f}}-\frac{6 i b^3 \sqrt{e} m n^3 \text{Li}_2\left (-\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{f}}+\frac{6 i b^3 \sqrt{e} m n^3 \text{Li}_2\left (\frac{i \sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{f}}+\frac{6 b^3 \sqrt{-e} m n^3 \text{Li}_3\left (-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{f}}-\frac{6 b^2 \sqrt{-e} m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{f}}-\frac{6 b^3 \sqrt{-e} m n^3 \text{Li}_3\left (\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{f}}+\frac{6 b^2 \sqrt{-e} m n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{f}}+\frac{6 b^3 \sqrt{-e} m n^3 \text{Li}_4\left (-\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{f}}-\frac{6 b^3 \sqrt{-e} m n^3 \text{Li}_4\left (\frac{\sqrt{f} x}{\sqrt{-e}}\right )}{\sqrt{f}}\\ \end{align*}
Mathematica [B] time = 0.698033, size = 2302, normalized size = 2.36 \[ \text{Result too large to show} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 28.803, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{3}\ln \left ( d \left ( f{x}^{2}+e \right ) ^{m} \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b^{3} \log \left (c x^{n}\right )^{3} + 3 \, a b^{2} \log \left (c x^{n}\right )^{2} + 3 \, a^{2} b \log \left (c x^{n}\right ) + a^{3}\right )} \log \left ({\left (f x^{2} + e\right )}^{m} d\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]