Optimal. Leaf size=630 \[ -\frac {16 a b e m n x}{9 f}+\frac {52 b^2 e m n^2 x}{27 f}-\frac {4}{27} b^2 m n^2 x^3-\frac {4 b^2 e^{3/2} m n^2 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{27 f^{3/2}}-\frac {16 b^2 e m n x \log \left (c x^n\right )}{9 f}+\frac {8}{27} b m n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {4 b e^{3/2} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 f^{3/2}}+\frac {2 e m x \left (a+b \log \left (c x^n\right )\right )^2}{3 f}-\frac {2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )^2-\frac {(-e)^{3/2} m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 f^{3/2}}+\frac {(-e)^{3/2} m \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 f^{3/2}}+\frac {2}{27} b^2 n^2 x^3 \log \left (d \left (e+f x^2\right )^m\right )-\frac {2}{9} b n x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^2\right )^m\right )+\frac {2 b (-e)^{3/2} m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 f^{3/2}}-\frac {2 b (-e)^{3/2} m n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 f^{3/2}}-\frac {2 i b^2 e^{3/2} m n^2 \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{9 f^{3/2}}+\frac {2 i b^2 e^{3/2} m n^2 \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{9 f^{3/2}}-\frac {2 b^2 (-e)^{3/2} m n^2 \text {Li}_3\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 f^{3/2}}+\frac {2 b^2 (-e)^{3/2} m n^2 \text {Li}_3\left (\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 f^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.69, 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 = {2342, 2341,
2425, 308, 211, 2393, 2332, 2361, 12, 4940, 2438, 2395, 2333, 2367, 2354, 2421, 6724}
\begin {gather*} \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 {4 b e^{3/2} m n \text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 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 {(-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 {4 b^2 e^{3/2} m n^2 \text {ArcTan}\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 {2}{27} b^2 n^2 x^3 \log \left (d \left (e+f x^2\right )^m\right )+\frac {52 b^2 e m n^2 x}{27 f}-\frac {4}{27} b^2 m n^2 x^3 \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 211
Rule 308
Rule 2332
Rule 2333
Rule 2341
Rule 2342
Rule 2354
Rule 2361
Rule 2367
Rule 2393
Rule 2395
Rule 2421
Rule 2425
Rule 2438
Rule 4940
Rule 6724
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 = 0.26, size = 1128, normalized size = 1.79 \begin {gather*} \frac {18 a^2 e \sqrt {f} m x-48 a b e \sqrt {f} m n x+52 b^2 e \sqrt {f} m n^2 x-6 a^2 f^{3/2} m x^3+8 a b f^{3/2} m n x^3-4 b^2 f^{3/2} m n^2 x^3-18 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 )-4 b^2 e^{3/2} m n^2 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )+36 a b e^{3/2} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log (x)-12 b^2 e^{3/2} m n^2 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log (x)-18 b^2 e^{3/2} m n^2 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log ^2(x)+36 a b e \sqrt {f} m x \log \left (c x^n\right )-48 b^2 e \sqrt {f} m n x \log \left (c x^n\right )-12 a b f^{3/2} m x^3 \log \left (c x^n\right )+8 b^2 f^{3/2} m n x^3 \log \left (c x^n\right )-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 )+18 b^2 e \sqrt {f} m x \log ^2\left (c x^n\right )-6 b^2 f^{3/2} m x^3 \log ^2\left (c x^n\right )-18 b^2 e^{3/2} m \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log ^2\left (c x^n\right )-18 i a b e^{3/2} m n \log (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 )+9 i b^2 e^{3/2} m n^2 \log ^2(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 )+18 i a b e^{3/2} m n \log (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 )-9 i b^2 e^{3/2} m n^2 \log ^2(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 a^2 f^{3/2} x^3 \log \left (d \left (e+f x^2\right )^m\right )-6 a b f^{3/2} n x^3 \log \left (d \left (e+f x^2\right )^m\right )+2 b^2 f^{3/2} n^2 x^3 \log \left (d \left (e+f x^2\right )^m\right )+18 a b f^{3/2} x^3 \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-6 b^2 f^{3/2} n x^3 \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+9 b^2 f^{3/2} x^3 \log ^2\left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\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}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int x^{2} \left (a +b \ln \left (c \,x^{n}\right )\right )^{2} \ln \left (d \left (f \,x^{2}+e \right )^{m}\right )\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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