Optimal. Leaf size=630 \[ \frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )-\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac {4 b e^{3/2} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 f^{3/2}}+\frac {2 b (-e)^{3/2} m n \text {Li}_2\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^{3/2}}-\frac {2 b (-e)^{3/2} m n \text {Li}_2\left (\frac {\sqrt {f} x}{\sqrt {-e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^{3/2}}-\frac {(-e)^{3/2} m \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{3 f^{3/2}}+\frac {(-e)^{3/2} m \log \left (\frac {\sqrt {f} x}{\sqrt {-e}}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{3 f^{3/2}}+\frac {2 e m x \left (a+b \log \left (c x^n\right )\right )^2}{3 f}-\frac {2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac {8}{27} b m n x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {16 a b e m n x}{9 f}-\frac {16 b^2 e m n x \log \left (c x^n\right )}{9 f}+\frac {2}{27} b^2 n^2 x^3 \log \left (d \left (e+f x^2\right )^m\right )-\frac {2 i b^2 e^{3/2} m n^2 \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{9 f^{3/2}}+\frac {2 i b^2 e^{3/2} m n^2 \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{9 f^{3/2}}-\frac {4 b^2 e^{3/2} m n^2 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{27 f^{3/2}}-\frac {2 b^2 (-e)^{3/2} m n^2 \text {Li}_3\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 f^{3/2}}+\frac {2 b^2 (-e)^{3/2} m n^2 \text {Li}_3\left (\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 f^{3/2}}+\frac {52 b^2 e m n^2 x}{27 f}-\frac {4}{27} b^2 m n^2 x^3 \]
[Out]
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Rubi [A] time = 1.07, antiderivative size = 630, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 17, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.607, Rules used = {2305, 2304, 2378, 302, 205, 2351, 2295, 2324, 12, 4848, 2391, 2353, 2296, 2330, 2317, 2374, 6589} \[ \frac {2 b (-e)^{3/2} m n \text {PolyLog}\left (2,-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^{3/2}}-\frac {2 b (-e)^{3/2} m n \text {PolyLog}\left (2,\frac {\sqrt {f} x}{\sqrt {-e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^{3/2}}-\frac {2 i b^2 e^{3/2} m n^2 \text {PolyLog}\left (2,-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{9 f^{3/2}}+\frac {2 i b^2 e^{3/2} m n^2 \text {PolyLog}\left (2,\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{9 f^{3/2}}-\frac {2 b^2 (-e)^{3/2} m n^2 \text {PolyLog}\left (3,-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 f^{3/2}}+\frac {2 b^2 (-e)^{3/2} m n^2 \text {PolyLog}\left (3,\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 f^{3/2}}+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )-\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac {4 b e^{3/2} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 f^{3/2}}-\frac {(-e)^{3/2} m \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{3 f^{3/2}}+\frac {(-e)^{3/2} m \log \left (\frac {\sqrt {f} x}{\sqrt {-e}}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{3 f^{3/2}}+\frac {2 e m x \left (a+b \log \left (c x^n\right )\right )^2}{3 f}-\frac {2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac {8}{27} b m n x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {16 a b e m n x}{9 f}-\frac {16 b^2 e m n x \log \left (c x^n\right )}{9 f}+\frac {2}{27} b^2 n^2 x^3 \log \left (d \left (e+f x^2\right )^m\right )-\frac {4 b^2 e^{3/2} m n^2 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{27 f^{3/2}}+\frac {52 b^2 e m n^2 x}{27 f}-\frac {4}{27} b^2 m n^2 x^3 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 205
Rule 302
Rule 2295
Rule 2296
Rule 2304
Rule 2305
Rule 2317
Rule 2324
Rule 2330
Rule 2351
Rule 2353
Rule 2374
Rule 2378
Rule 2391
Rule 4848
Rule 6589
Rubi steps
\begin {align*} \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right ) \, dx &=\frac {2}{27} b^2 n^2 x^3 \log \left (d \left (e+f x^2\right )^m\right )-\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )-(2 f m) \int \left (\frac {2 b^2 n^2 x^4}{27 \left (e+f x^2\right )}-\frac {2 b n x^4 \left (a+b \log \left (c x^n\right )\right )}{9 \left (e+f x^2\right )}+\frac {x^4 \left (a+b \log \left (c x^n\right )\right )^2}{3 \left (e+f x^2\right )}\right ) \, dx\\ &=\frac {2}{27} b^2 n^2 x^3 \log \left (d \left (e+f x^2\right )^m\right )-\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )-\frac {1}{3} (2 f m) \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )^2}{e+f x^2} \, dx+\frac {1}{9} (4 b f m n) \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{e+f x^2} \, dx-\frac {1}{27} \left (4 b^2 f m n^2\right ) \int \frac {x^4}{e+f x^2} \, dx\\ &=\frac {2}{27} b^2 n^2 x^3 \log \left (d \left (e+f x^2\right )^m\right )-\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )-\frac {1}{3} (2 f m) \int \left (-\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{f^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{f}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^2}{f^2 \left (e+f x^2\right )}\right ) \, dx+\frac {1}{9} (4 b f m n) \int \left (-\frac {e \left (a+b \log \left (c x^n\right )\right )}{f^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{f}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{f^2 \left (e+f x^2\right )}\right ) \, dx-\frac {1}{27} \left (4 b^2 f m n^2\right ) \int \left (-\frac {e}{f^2}+\frac {x^2}{f}+\frac {e^2}{f^2 \left (e+f x^2\right )}\right ) \, dx\\ &=\frac {4 b^2 e m n^2 x}{27 f}-\frac {4}{81} b^2 m n^2 x^3+\frac {2}{27} b^2 n^2 x^3 \log \left (d \left (e+f x^2\right )^m\right )-\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )-\frac {1}{3} (2 m) \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx+\frac {(2 e m) \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{3 f}-\frac {\left (2 e^2 m\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{e+f x^2} \, dx}{3 f}+\frac {1}{9} (4 b m n) \int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx-\frac {(4 b e m n) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{9 f}+\frac {\left (4 b e^2 m n\right ) \int \frac {a+b \log \left (c x^n\right )}{e+f x^2} \, dx}{9 f}-\frac {\left (4 b^2 e^2 m n^2\right ) \int \frac {1}{e+f x^2} \, dx}{27 f}\\ &=-\frac {4 a b e m n x}{9 f}+\frac {4 b^2 e m n^2 x}{27 f}-\frac {8}{81} b^2 m n^2 x^3-\frac {4 b^2 e^{3/2} m n^2 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{27 f^{3/2}}+\frac {4}{27} b m n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {4 b e^{3/2} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 f^{3/2}}+\frac {2 e m x \left (a+b \log \left (c x^n\right )\right )^2}{3 f}-\frac {2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac {2}{27} b^2 n^2 x^3 \log \left (d \left (e+f x^2\right )^m\right )-\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )-\frac {\left (2 e^2 m\right ) \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}{3 f}+\frac {1}{9} (4 b m n) \int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx-\frac {(4 b e m n) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{3 f}-\frac {\left (4 b^2 e m n\right ) \int \log \left (c x^n\right ) \, dx}{9 f}-\frac {\left (4 b^2 e^2 m n^2\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} \sqrt {f} x} \, dx}{9 f}\\ &=-\frac {16 a b e m n x}{9 f}+\frac {16 b^2 e m n^2 x}{27 f}-\frac {4}{27} b^2 m n^2 x^3-\frac {4 b^2 e^{3/2} m n^2 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{27 f^{3/2}}-\frac {4 b^2 e m n x \log \left (c x^n\right )}{9 f}+\frac {8}{27} b m n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {4 b e^{3/2} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 f^{3/2}}+\frac {2 e m x \left (a+b \log \left (c x^n\right )\right )^2}{3 f}-\frac {2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac {2}{27} b^2 n^2 x^3 \log \left (d \left (e+f x^2\right )^m\right )-\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac {\left ((-e)^{3/2} m\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {-e}-\sqrt {f} x} \, dx}{3 f}+\frac {\left ((-e)^{3/2} m\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {-e}+\sqrt {f} x} \, dx}{3 f}-\frac {\left (4 b^2 e m n\right ) \int \log \left (c x^n\right ) \, dx}{3 f}-\frac {\left (4 b^2 e^{3/2} m n^2\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{x} \, dx}{9 f^{3/2}}\\ &=-\frac {16 a b e m n x}{9 f}+\frac {52 b^2 e m n^2 x}{27 f}-\frac {4}{27} b^2 m n^2 x^3-\frac {4 b^2 e^{3/2} m n^2 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{27 f^{3/2}}-\frac {16 b^2 e m n x \log \left (c x^n\right )}{9 f}+\frac {8}{27} b m n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {4 b e^{3/2} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 f^{3/2}}+\frac {2 e m x \left (a+b \log \left (c x^n\right )\right )^2}{3 f}-\frac {2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac {(-e)^{3/2} m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 f^{3/2}}+\frac {(-e)^{3/2} m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 f^{3/2}}+\frac {2}{27} b^2 n^2 x^3 \log \left (d \left (e+f x^2\right )^m\right )-\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac {\left (2 b (-e)^{3/2} m n\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}{3 f^{3/2}}-\frac {\left (2 b (-e)^{3/2} m n\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}{3 f^{3/2}}-\frac {\left (2 i b^2 e^{3/2} m n^2\right ) \int \frac {\log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{x} \, dx}{9 f^{3/2}}+\frac {\left (2 i b^2 e^{3/2} m n^2\right ) \int \frac {\log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{x} \, dx}{9 f^{3/2}}\\ &=-\frac {16 a b e m n x}{9 f}+\frac {52 b^2 e m n^2 x}{27 f}-\frac {4}{27} b^2 m n^2 x^3-\frac {4 b^2 e^{3/2} m n^2 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{27 f^{3/2}}-\frac {16 b^2 e m n x \log \left (c x^n\right )}{9 f}+\frac {8}{27} b m n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {4 b e^{3/2} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 f^{3/2}}+\frac {2 e m x \left (a+b \log \left (c x^n\right )\right )^2}{3 f}-\frac {2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac {(-e)^{3/2} m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 f^{3/2}}+\frac {(-e)^{3/2} m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 f^{3/2}}+\frac {2}{27} b^2 n^2 x^3 \log \left (d \left (e+f x^2\right )^m\right )-\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac {2 b (-e)^{3/2} m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 f^{3/2}}-\frac {2 b (-e)^{3/2} m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 f^{3/2}}-\frac {2 i b^2 e^{3/2} m n^2 \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{9 f^{3/2}}+\frac {2 i b^2 e^{3/2} m n^2 \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{9 f^{3/2}}-\frac {\left (2 b^2 (-e)^{3/2} m n^2\right ) \int \frac {\text {Li}_2\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{x} \, dx}{3 f^{3/2}}+\frac {\left (2 b^2 (-e)^{3/2} m n^2\right ) \int \frac {\text {Li}_2\left (\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{x} \, dx}{3 f^{3/2}}\\ &=-\frac {16 a b e m n x}{9 f}+\frac {52 b^2 e m n^2 x}{27 f}-\frac {4}{27} b^2 m n^2 x^3-\frac {4 b^2 e^{3/2} m n^2 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{27 f^{3/2}}-\frac {16 b^2 e m n x \log \left (c x^n\right )}{9 f}+\frac {8}{27} b m n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {4 b e^{3/2} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 f^{3/2}}+\frac {2 e m x \left (a+b \log \left (c x^n\right )\right )^2}{3 f}-\frac {2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac {(-e)^{3/2} m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 f^{3/2}}+\frac {(-e)^{3/2} m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 f^{3/2}}+\frac {2}{27} b^2 n^2 x^3 \log \left (d \left (e+f x^2\right )^m\right )-\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac {2 b (-e)^{3/2} m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 f^{3/2}}-\frac {2 b (-e)^{3/2} m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 f^{3/2}}-\frac {2 i b^2 e^{3/2} m n^2 \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{9 f^{3/2}}+\frac {2 i b^2 e^{3/2} m n^2 \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{9 f^{3/2}}-\frac {2 b^2 (-e)^{3/2} m n^2 \text {Li}_3\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 f^{3/2}}+\frac {2 b^2 (-e)^{3/2} m n^2 \text {Li}_3\left (\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 f^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.45, size = 1128, normalized size = 1.79 \[ \frac {-4 b^2 f^{3/2} m n^2 x^3-6 b^2 f^{3/2} m \log ^2\left (c x^n\right ) x^3-6 a^2 f^{3/2} m x^3+8 a b f^{3/2} m n x^3-12 a b f^{3/2} m \log \left (c x^n\right ) x^3+8 b^2 f^{3/2} m n \log \left (c x^n\right ) x^3+2 b^2 f^{3/2} n^2 \log \left (d \left (f x^2+e\right )^m\right ) x^3+9 b^2 f^{3/2} \log ^2\left (c x^n\right ) \log \left (d \left (f x^2+e\right )^m\right ) x^3+9 a^2 f^{3/2} \log \left (d \left (f x^2+e\right )^m\right ) x^3-6 a b f^{3/2} n \log \left (d \left (f x^2+e\right )^m\right ) x^3+18 a b f^{3/2} \log \left (c x^n\right ) \log \left (d \left (f x^2+e\right )^m\right ) x^3-6 b^2 f^{3/2} n \log \left (c x^n\right ) \log \left (d \left (f x^2+e\right )^m\right ) x^3+52 b^2 e \sqrt {f} m n^2 x+18 b^2 e \sqrt {f} m \log ^2\left (c x^n\right ) x+18 a^2 e \sqrt {f} m x-48 a b e \sqrt {f} m n x+36 a b e \sqrt {f} m \log \left (c x^n\right ) x-48 b^2 e \sqrt {f} m n \log \left (c x^n\right ) x-18 b^2 e^{3/2} m n^2 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log ^2(x)-18 b^2 e^{3/2} m \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log ^2\left (c x^n\right )-4 b^2 e^{3/2} m n^2 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )-18 a^2 e^{3/2} m \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )+12 a b e^{3/2} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )-12 b^2 e^{3/2} m n^2 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log (x)+36 a b e^{3/2} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log (x)-36 a b e^{3/2} m \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log \left (c x^n\right )+12 b^2 e^{3/2} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log \left (c x^n\right )+36 b^2 e^{3/2} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log (x) \log \left (c x^n\right )+9 i b^2 e^{3/2} m n^2 \log ^2(x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+6 i b^2 e^{3/2} m n^2 \log (x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-18 i a b e^{3/2} m n \log (x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-18 i b^2 e^{3/2} m n \log (x) \log \left (c x^n\right ) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-9 i b^2 e^{3/2} m n^2 \log ^2(x) \log \left (\frac {i \sqrt {f} x}{\sqrt {e}}+1\right )-6 i b^2 e^{3/2} m n^2 \log (x) \log \left (\frac {i \sqrt {f} x}{\sqrt {e}}+1\right )+18 i a b e^{3/2} m n \log (x) \log \left (\frac {i \sqrt {f} x}{\sqrt {e}}+1\right )+18 i b^2 e^{3/2} m n \log (x) \log \left (c x^n\right ) \log \left (\frac {i \sqrt {f} x}{\sqrt {e}}+1\right )+6 i b e^{3/2} m n \left (3 a-b n+3 b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+6 i b e^{3/2} m n \left (-3 a+b n-3 b \log \left (c x^n\right )\right ) \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )-18 i b^2 e^{3/2} m n^2 \text {Li}_3\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+18 i b^2 e^{3/2} m n^2 \text {Li}_3\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{27 f^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.05, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{2} x^{2} \log \left (c x^{n}\right )^{2} + 2 \, a b x^{2} \log \left (c x^{n}\right ) + a^{2} x^{2}\right )} \log \left ({\left (f x^{2} + e\right )}^{m} d\right ), x\right ) \]
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 \[ \int {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x^{2} \log \left ({\left (f x^{2} + e\right )}^{m} d\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 111.55, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \,x^{n}\right )+a \right )^{2} x^{2} \ln \left (d \left (f \,x^{2}+e \right )^{m}\right )\, dx \]
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 \[ \frac {1}{27} \, {\left (9 \, b^{2} m x^{3} \log \left (x^{n}\right )^{2} - 6 \, {\left ({\left (m n - 3 \, m \log \relax (c)\right )} b^{2} - 3 \, a b m\right )} x^{3} \log \left (x^{n}\right ) - {\left (6 \, {\left (m n - 3 \, m \log \relax (c)\right )} a b - {\left (2 \, m n^{2} - 6 \, m n \log \relax (c) + 9 \, m \log \relax (c)^{2}\right )} b^{2} - 9 \, a^{2} m\right )} x^{3}\right )} \log \left (f x^{2} + e\right ) + \int -\frac {{\left (9 \, {\left (2 \, f m - 3 \, f \log \relax (d)\right )} a^{2} - 6 \, {\left (2 \, f m n - 3 \, {\left (2 \, f m - 3 \, f \log \relax (d)\right )} \log \relax (c)\right )} a b + {\left (4 \, f m n^{2} - 12 \, f m n \log \relax (c) + 9 \, {\left (2 \, f m - 3 \, f \log \relax (d)\right )} \log \relax (c)^{2}\right )} b^{2}\right )} x^{4} - 27 \, {\left (b^{2} e \log \relax (c)^{2} \log \relax (d) + 2 \, a b e \log \relax (c) \log \relax (d) + a^{2} e \log \relax (d)\right )} x^{2} + 9 \, {\left ({\left (2 \, f m - 3 \, f \log \relax (d)\right )} b^{2} x^{4} - 3 \, b^{2} e x^{2} \log \relax (d)\right )} \log \left (x^{n}\right )^{2} + 6 \, {\left ({\left (3 \, {\left (2 \, f m - 3 \, f \log \relax (d)\right )} a b - {\left (2 \, f m n - 3 \, {\left (2 \, f m - 3 \, f \log \relax (d)\right )} \log \relax (c)\right )} b^{2}\right )} x^{4} - 9 \, {\left (b^{2} e \log \relax (c) \log \relax (d) + a b e \log \relax (d)\right )} x^{2}\right )} \log \left (x^{n}\right )}{27 \, {\left (f x^{2} + e\right )}}\,{d x} \]
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 \[ \int x^2\,\ln \left (d\,{\left (f\,x^2+e\right )}^m\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2 \,d x \]
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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