Optimal. Leaf size=977 \[ 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 {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right ) b^3}{\sqrt {f}}+\frac {6 i \sqrt {e} m n^3 \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right ) b^3}{\sqrt {f}}+\frac {6 \sqrt {-e} m n^3 \text {Li}_3\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) b^3}{\sqrt {f}}-\frac {6 \sqrt {-e} m n^3 \text {Li}_3\left (\frac {\sqrt {f} x}{\sqrt {-e}}\right ) b^3}{\sqrt {f}}+\frac {6 \sqrt {-e} m n^3 \text {Li}_4\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) b^3}{\sqrt {f}}-\frac {6 \sqrt {-e} m n^3 \text {Li}_4\left (\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 {Li}_2\left (-\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 {Li}_2\left (\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 {Li}_3\left (-\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 {Li}_3\left (\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 {Li}_2\left (-\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 {Li}_2\left (\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 ) \]
[Out]
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Rubi [A] time = 1.50, antiderivative size = 977, normalized size of antiderivative = 1.00, number of steps used = 42, number of rules used = 17, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.680, 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 6
Rule 12
Rule 205
Rule 321
Rule 2295
Rule 2296
Rule 2317
Rule 2324
Rule 2330
Rule 2351
Rule 2353
Rule 2371
Rule 2374
Rule 2383
Rule 2391
Rule 4848
Rule 6589
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*}
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Mathematica [B] time = 0.75, size = 2302, normalized size = 2.36 \[ \text {Result too large to show} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{3} \log \left (c x^{n}\right )^{3} + 3 \, a b^{2} \log \left (c x^{n}\right )^{2} + 3 \, a^{2} b \log \left (c x^{n}\right ) + a^{3}\right )} \log \left ({\left (f x^{2} + e\right )}^{m} d\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 123.98, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \,x^{n}\right )+a \right )^{3} \ln \left (d \left (f \,x^{2}+e \right )^{m}\right )\, dx \]
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 \[ {\left (b^{3} m x \log \left (x^{n}\right )^{3} - 3 \, {\left ({\left (m n - m \log \relax (c)\right )} b^{3} - a b^{2} m\right )} x \log \left (x^{n}\right )^{2} - 3 \, {\left (2 \, {\left (m n - m \log \relax (c)\right )} a b^{2} - {\left (2 \, m n^{2} - 2 \, m n \log \relax (c) + m \log \relax (c)^{2}\right )} b^{3} - a^{2} b m\right )} x \log \left (x^{n}\right ) - {\left (3 \, {\left (m n - m \log \relax (c)\right )} a^{2} b - 3 \, {\left (2 \, m n^{2} - 2 \, m n \log \relax (c) + m \log \relax (c)^{2}\right )} a b^{2} + {\left (6 \, m n^{3} - 6 \, m n^{2} \log \relax (c) + 3 \, m n \log \relax (c)^{2} - m \log \relax (c)^{3}\right )} b^{3} - a^{3} m\right )} x\right )} \log \left (f x^{2} + e\right ) + \int \frac {b^{3} e \log \relax (c)^{3} \log \relax (d) + 3 \, a b^{2} e \log \relax (c)^{2} \log \relax (d) + 3 \, a^{2} b e \log \relax (c) \log \relax (d) + a^{3} e \log \relax (d) - {\left ({\left (2 \, f m - f \log \relax (d)\right )} b^{3} x^{2} - b^{3} e \log \relax (d)\right )} \log \left (x^{n}\right )^{3} - {\left ({\left (2 \, f m - f \log \relax (d)\right )} a^{3} - 3 \, {\left (2 \, f m n - {\left (2 \, f m - f \log \relax (d)\right )} \log \relax (c)\right )} a^{2} b + 3 \, {\left (4 \, f m n^{2} - 4 \, f m n \log \relax (c) + {\left (2 \, f m - f \log \relax (d)\right )} \log \relax (c)^{2}\right )} a b^{2} - {\left (12 \, f m n^{3} - 12 \, f m n^{2} \log \relax (c) + 6 \, f m n \log \relax (c)^{2} - {\left (2 \, f m - f \log \relax (d)\right )} \log \relax (c)^{3}\right )} b^{3}\right )} x^{2} + 3 \, {\left (b^{3} e \log \relax (c) \log \relax (d) + a b^{2} e \log \relax (d) - {\left ({\left (2 \, f m - f \log \relax (d)\right )} a b^{2} - {\left (2 \, f m n - {\left (2 \, f m - f \log \relax (d)\right )} \log \relax (c)\right )} b^{3}\right )} x^{2}\right )} \log \left (x^{n}\right )^{2} + 3 \, {\left (b^{3} e \log \relax (c)^{2} \log \relax (d) + 2 \, a b^{2} e \log \relax (c) \log \relax (d) + a^{2} b e \log \relax (d) - {\left ({\left (2 \, f m - f \log \relax (d)\right )} a^{2} b - 2 \, {\left (2 \, f m n - {\left (2 \, f m - f \log \relax (d)\right )} \log \relax (c)\right )} a b^{2} + {\left (4 \, f m n^{2} - 4 \, f m n \log \relax (c) + {\left (2 \, f m - f \log \relax (d)\right )} \log \relax (c)^{2}\right )} b^{3}\right )} x^{2}\right )} \log \left (x^{n}\right )}{f x^{2} + e}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \ln \left (d\,{\left (f\,x^2+e\right )}^m\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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