Optimal. Leaf size=1092 \[ \frac {52 a b^2 e m n^2 x}{9 f}-\frac {160 b^3 e m n^3 x}{27 f}+\frac {16}{81} b^3 m n^3 x^3+\frac {4 b^3 e^{3/2} m n^3 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{27 f^{3/2}}+\frac {52 b^3 e m n^2 x \log \left (c x^n\right )}{9 f}-\frac {4}{9} b^2 m n^2 x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {4 b^2 e^{3/2} m n^2 \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 {8 b e m n x \left (a+b \log \left (c x^n\right )\right )^2}{3 f}+\frac {4}{9} b m n x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac {2 e m x \left (a+b \log \left (c x^n\right )\right )^3}{3 f}-\frac {2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )^3+\frac {b (-e)^{3/2} m n \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 )^3 \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 f^{3/2}}-\frac {b (-e)^{3/2} m n \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 )^3 \log \left (1+\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 f^{3/2}}-\frac {2}{27} b^3 n^3 x^3 \log \left (d \left (e+f x^2\right )^m\right )+\frac {2}{9} b^2 n^2 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} b n 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} x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )-\frac {2 b^2 (-e)^{3/2} m n^2 \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 {b (-e)^{3/2} m n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{f^{3/2}}+\frac {2 b^2 (-e)^{3/2} m n^2 \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 {b (-e)^{3/2} m n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{f^{3/2}}+\frac {2 i b^3 e^{3/2} m n^3 \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{9 f^{3/2}}-\frac {2 i b^3 e^{3/2} m n^3 \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{9 f^{3/2}}+\frac {2 b^3 (-e)^{3/2} m n^3 \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 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{f^{3/2}}-\frac {2 b^3 (-e)^{3/2} m n^3 \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 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{f^{3/2}}+\frac {2 b^3 (-e)^{3/2} m n^3 \text {Li}_4\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{f^{3/2}}-\frac {2 b^3 (-e)^{3/2} m n^3 \text {Li}_4\left (\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{f^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 1.23, antiderivative size = 1092, normalized size of antiderivative = 1.00, number of steps
used = 49, number of rules used = 18, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {2342, 2341,
2425, 308, 211, 2393, 2332, 2361, 12, 4940, 2438, 2395, 2333, 2367, 2354, 2421, 6724, 2430}
\begin {gather*} \frac {16}{81} m n^3 x^3 b^3-\frac {160 e m n^3 x b^3}{27 f}+\frac {4 e^{3/2} m n^3 \text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) b^3}{27 f^{3/2}}+\frac {52 e m n^2 x \log \left (c x^n\right ) b^3}{9 f}-\frac {2}{27} n^3 x^3 \log \left (d \left (f x^2+e\right )^m\right ) b^3+\frac {2 i e^{3/2} m n^3 \text {PolyLog}\left (2,-\frac {i \sqrt {f} x}{\sqrt {e}}\right ) b^3}{9 f^{3/2}}-\frac {2 i e^{3/2} m n^3 \text {PolyLog}\left (2,\frac {i \sqrt {f} x}{\sqrt {e}}\right ) b^3}{9 f^{3/2}}+\frac {2 (-e)^{3/2} m n^3 \text {PolyLog}\left (3,-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) b^3}{3 f^{3/2}}-\frac {2 (-e)^{3/2} m n^3 \text {PolyLog}\left (3,\frac {\sqrt {f} x}{\sqrt {-e}}\right ) b^3}{3 f^{3/2}}+\frac {2 (-e)^{3/2} m n^3 \text {PolyLog}\left (4,-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) b^3}{f^{3/2}}-\frac {2 (-e)^{3/2} m n^3 \text {PolyLog}\left (4,\frac {\sqrt {f} x}{\sqrt {-e}}\right ) b^3}{f^{3/2}}+\frac {52 a e m n^2 x b^2}{9 f}-\frac {4}{9} m n^2 x^3 \left (a+b \log \left (c x^n\right )\right ) b^2-\frac {4 e^{3/2} m n^2 \text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right ) b^2}{9 f^{3/2}}+\frac {2}{9} n^2 x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (f x^2+e\right )^m\right ) b^2-\frac {2 (-e)^{3/2} 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}{3 f^{3/2}}+\frac {2 (-e)^{3/2} 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}{3 f^{3/2}}-\frac {2 (-e)^{3/2} 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}{f^{3/2}}+\frac {2 (-e)^{3/2} 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}{f^{3/2}}+\frac {4}{9} m n x^3 \left (a+b \log \left (c x^n\right )\right )^2 b-\frac {8 e m n x \left (a+b \log \left (c x^n\right )\right )^2 b}{3 f}+\frac {(-e)^{3/2} m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right ) b}{3 f^{3/2}}-\frac {(-e)^{3/2} m n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (\frac {\sqrt {f} x}{\sqrt {-e}}+1\right ) b}{3 f^{3/2}}-\frac {1}{3} n x^3 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (f x^2+e\right )^m\right ) b+\frac {(-e)^{3/2} 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}{f^{3/2}}-\frac {(-e)^{3/2} 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}{f^{3/2}}-\frac {2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )^3+\frac {2 e m x \left (a+b \log \left (c x^n\right )\right )^3}{3 f}-\frac {(-e)^{3/2} m \left (a+b \log \left (c x^n\right )\right )^3 \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 )^3 \log \left (\frac {\sqrt {f} x}{\sqrt {-e}}+1\right )}{3 f^{3/2}}+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (f x^2+e\right )^m\right ) \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 2430
Rule 2438
Rule 4940
Rule 6724
Rubi steps
\begin {align*} \int x^2 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right ) \, dx &=-\frac {2}{27} b^3 n^3 x^3 \log \left (d \left (e+f x^2\right )^m\right )+\frac {2}{9} b^2 n^2 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} b n 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} x^3 \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 {2 b^3 n^3 x^4}{27 \left (e+f x^2\right )}+\frac {2 b^2 n^2 x^4 \left (a+b \log \left (c x^n\right )\right )}{9 \left (e+f x^2\right )}-\frac {b n x^4 \left (a+b \log \left (c x^n\right )\right )^2}{3 \left (e+f x^2\right )}+\frac {x^4 \left (a+b \log \left (c x^n\right )\right )^3}{3 \left (e+f x^2\right )}\right ) \, dx\\ &=-\frac {2}{27} b^3 n^3 x^3 \log \left (d \left (e+f x^2\right )^m\right )+\frac {2}{9} b^2 n^2 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} b n 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} x^3 \left (a+b \log \left (c x^n\right )\right )^3 \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 )^3}{e+f x^2} \, dx+\frac {1}{3} (2 b f m n) \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )^2}{e+f x^2} \, dx-\frac {1}{9} \left (4 b^2 f m n^2\right ) \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^3 f m n^3\right ) \int \frac {x^4}{e+f x^2} \, dx\\ &=-\frac {2}{27} b^3 n^3 x^3 \log \left (d \left (e+f x^2\right )^m\right )+\frac {2}{9} b^2 n^2 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} b n 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} x^3 \left (a+b \log \left (c x^n\right )\right )^3 \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 )^3}{f^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^3}{f}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )^3}{f^2 \left (e+f x^2\right )}\right ) \, dx+\frac {1}{3} (2 b f m n) \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} \left (4 b^2 f m n^2\right ) \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^3 f m n^3\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^3 e m n^3 x}{27 f}+\frac {4}{81} b^3 m n^3 x^3-\frac {2}{27} b^3 n^3 x^3 \log \left (d \left (e+f x^2\right )^m\right )+\frac {2}{9} b^2 n^2 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} b n 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} x^3 \left (a+b \log \left (c x^n\right )\right )^3 \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 )^3 \, dx+\frac {(2 e m) \int \left (a+b \log \left (c x^n\right )\right )^3 \, dx}{3 f}-\frac {\left (2 e^2 m\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{e+f x^2} \, dx}{3 f}+\frac {1}{3} (2 b m n) \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx-\frac {(2 b e m n) \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{3 f}+\frac {\left (2 b e^2 m n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{e+f x^2} \, dx}{3 f}-\frac {1}{9} \left (4 b^2 m n^2\right ) \int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx+\frac {\left (4 b^2 e m n^2\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{9 f}-\frac {\left (4 b^2 e^2 m n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{e+f x^2} \, dx}{9 f}+\frac {\left (4 b^3 e^2 m n^3\right ) \int \frac {1}{e+f x^2} \, dx}{27 f}\\ &=\frac {4 a b^2 e m n^2 x}{9 f}-\frac {4 b^3 e m n^3 x}{27 f}+\frac {8}{81} b^3 m n^3 x^3+\frac {4 b^3 e^{3/2} m n^3 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{27 f^{3/2}}-\frac {4}{27} b^2 m n^2 x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {4 b^2 e^{3/2} m n^2 \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 m n x \left (a+b \log \left (c x^n\right )\right )^2}{3 f}+\frac {2}{9} b m n x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac {2 e m x \left (a+b \log \left (c x^n\right )\right )^3}{3 f}-\frac {2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )^3-\frac {2}{27} b^3 n^3 x^3 \log \left (d \left (e+f x^2\right )^m\right )+\frac {2}{9} b^2 n^2 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} b n 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} x^3 \left (a+b \log \left (c x^n\right )\right )^3 \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 )^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}{3 f}+\frac {1}{3} (2 b m n) \int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx-\frac {(2 b e m n) \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{f}+\frac {\left (2 b e^2 m n\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} \left (4 b^2 m n^2\right ) \int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx+\frac {\left (4 b^2 e m n^2\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{3 f}+\frac {\left (4 b^3 e m n^2\right ) \int \log \left (c x^n\right ) \, dx}{9 f}+\frac {\left (4 b^3 e^2 m n^3\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^2 e m n^2 x}{9 f}-\frac {16 b^3 e m n^3 x}{27 f}+\frac {4}{27} b^3 m n^3 x^3+\frac {4 b^3 e^{3/2} m n^3 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{27 f^{3/2}}+\frac {4 b^3 e m n^2 x \log \left (c x^n\right )}{9 f}-\frac {8}{27} b^2 m n^2 x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {4 b^2 e^{3/2} m n^2 \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 {8 b e m n x \left (a+b \log \left (c x^n\right )\right )^2}{3 f}+\frac {4}{9} b m n x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac {2 e m x \left (a+b \log \left (c x^n\right )\right )^3}{3 f}-\frac {2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )^3-\frac {2}{27} b^3 n^3 x^3 \log \left (d \left (e+f x^2\right )^m\right )+\frac {2}{9} b^2 n^2 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} b n 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} x^3 \left (a+b \log \left (c x^n\right )\right )^3 \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 )^3}{\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 )^3}{\sqrt {-e}+\sqrt {f} x} \, dx}{3 f}-\frac {\left (b (-e)^{3/2} m n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {-e}-\sqrt {f} x} \, dx}{3 f}-\frac {\left (b (-e)^{3/2} m n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {-e}+\sqrt {f} x} \, dx}{3 f}-\frac {1}{9} \left (4 b^2 m n^2\right ) \int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx+\frac {\left (4 b^2 e m n^2\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{f}+\frac {\left (4 b^3 e m n^2\right ) \int \log \left (c x^n\right ) \, dx}{3 f}+\frac {\left (4 b^3 e^{3/2} m n^3\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{x} \, dx}{9 f^{3/2}}\\ &=\frac {52 a b^2 e m n^2 x}{9 f}-\frac {52 b^3 e m n^3 x}{27 f}+\frac {16}{81} b^3 m n^3 x^3+\frac {4 b^3 e^{3/2} m n^3 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{27 f^{3/2}}+\frac {16 b^3 e m n^2 x \log \left (c x^n\right )}{9 f}-\frac {4}{9} b^2 m n^2 x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {4 b^2 e^{3/2} m n^2 \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 {8 b e m n x \left (a+b \log \left (c x^n\right )\right )^2}{3 f}+\frac {4}{9} b m n x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac {2 e m x \left (a+b \log \left (c x^n\right )\right )^3}{3 f}-\frac {2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )^3+\frac {b (-e)^{3/2} m n \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 )^3 \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 f^{3/2}}-\frac {b (-e)^{3/2} m n \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 )^3 \log \left (1+\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 f^{3/2}}-\frac {2}{27} b^3 n^3 x^3 \log \left (d \left (e+f x^2\right )^m\right )+\frac {2}{9} b^2 n^2 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} b n 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} x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )+\frac {\left (b (-e)^{3/2} 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}{f^{3/2}}-\frac {\left (b (-e)^{3/2} 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}{f^{3/2}}-\frac {\left (2 b^2 (-e)^{3/2} 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}{3 f^{3/2}}+\frac {\left (2 b^2 (-e)^{3/2} 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}{3 f^{3/2}}+\frac {\left (4 b^3 e m n^2\right ) \int \log \left (c x^n\right ) \, dx}{f}+\frac {\left (2 i b^3 e^{3/2} m n^3\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^3 e^{3/2} m n^3\right ) \int \frac {\log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{x} \, dx}{9 f^{3/2}}\\ &=\frac {52 a b^2 e m n^2 x}{9 f}-\frac {160 b^3 e m n^3 x}{27 f}+\frac {16}{81} b^3 m n^3 x^3+\frac {4 b^3 e^{3/2} m n^3 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{27 f^{3/2}}+\frac {52 b^3 e m n^2 x \log \left (c x^n\right )}{9 f}-\frac {4}{9} b^2 m n^2 x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {4 b^2 e^{3/2} m n^2 \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 {8 b e m n x \left (a+b \log \left (c x^n\right )\right )^2}{3 f}+\frac {4}{9} b m n x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac {2 e m x \left (a+b \log \left (c x^n\right )\right )^3}{3 f}-\frac {2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )^3+\frac {b (-e)^{3/2} m n \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 )^3 \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 f^{3/2}}-\frac {b (-e)^{3/2} m n \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 )^3 \log \left (1+\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 f^{3/2}}-\frac {2}{27} b^3 n^3 x^3 \log \left (d \left (e+f x^2\right )^m\right )+\frac {2}{9} b^2 n^2 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} b n 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} x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )-\frac {2 b^2 (-e)^{3/2} m n^2 \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 {b (-e)^{3/2} m n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{f^{3/2}}+\frac {2 b^2 (-e)^{3/2} m n^2 \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 {b (-e)^{3/2} m n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{f^{3/2}}+\frac {2 i b^3 e^{3/2} m n^3 \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{9 f^{3/2}}-\frac {2 i b^3 e^{3/2} m n^3 \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 {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{x} \, dx}{f^{3/2}}+\frac {\left (2 b^2 (-e)^{3/2} 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}{f^{3/2}}+\frac {\left (2 b^3 (-e)^{3/2} m n^3\right ) \int \frac {\text {Li}_2\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{x} \, dx}{3 f^{3/2}}-\frac {\left (2 b^3 (-e)^{3/2} m n^3\right ) \int \frac {\text {Li}_2\left (\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{x} \, dx}{3 f^{3/2}}\\ &=\frac {52 a b^2 e m n^2 x}{9 f}-\frac {160 b^3 e m n^3 x}{27 f}+\frac {16}{81} b^3 m n^3 x^3+\frac {4 b^3 e^{3/2} m n^3 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{27 f^{3/2}}+\frac {52 b^3 e m n^2 x \log \left (c x^n\right )}{9 f}-\frac {4}{9} b^2 m n^2 x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {4 b^2 e^{3/2} m n^2 \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 {8 b e m n x \left (a+b \log \left (c x^n\right )\right )^2}{3 f}+\frac {4}{9} b m n x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac {2 e m x \left (a+b \log \left (c x^n\right )\right )^3}{3 f}-\frac {2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )^3+\frac {b (-e)^{3/2} m n \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 )^3 \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 f^{3/2}}-\frac {b (-e)^{3/2} m n \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 )^3 \log \left (1+\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 f^{3/2}}-\frac {2}{27} b^3 n^3 x^3 \log \left (d \left (e+f x^2\right )^m\right )+\frac {2}{9} b^2 n^2 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} b n 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} x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )-\frac {2 b^2 (-e)^{3/2} m n^2 \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 {b (-e)^{3/2} m n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{f^{3/2}}+\frac {2 b^2 (-e)^{3/2} m n^2 \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 {b (-e)^{3/2} m n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{f^{3/2}}+\frac {2 i b^3 e^{3/2} m n^3 \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{9 f^{3/2}}-\frac {2 i b^3 e^{3/2} m n^3 \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{9 f^{3/2}}+\frac {2 b^3 (-e)^{3/2} m n^3 \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 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{f^{3/2}}-\frac {2 b^3 (-e)^{3/2} m n^3 \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 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{f^{3/2}}+\frac {\left (2 b^3 (-e)^{3/2} m n^3\right ) \int \frac {\text {Li}_3\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{x} \, dx}{f^{3/2}}-\frac {\left (2 b^3 (-e)^{3/2} m n^3\right ) \int \frac {\text {Li}_3\left (\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{x} \, dx}{f^{3/2}}\\ &=\frac {52 a b^2 e m n^2 x}{9 f}-\frac {160 b^3 e m n^3 x}{27 f}+\frac {16}{81} b^3 m n^3 x^3+\frac {4 b^3 e^{3/2} m n^3 \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{27 f^{3/2}}+\frac {52 b^3 e m n^2 x \log \left (c x^n\right )}{9 f}-\frac {4}{9} b^2 m n^2 x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {4 b^2 e^{3/2} m n^2 \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 {8 b e m n x \left (a+b \log \left (c x^n\right )\right )^2}{3 f}+\frac {4}{9} b m n x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac {2 e m x \left (a+b \log \left (c x^n\right )\right )^3}{3 f}-\frac {2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )^3+\frac {b (-e)^{3/2} m n \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 )^3 \log \left (1-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 f^{3/2}}-\frac {b (-e)^{3/2} m n \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 )^3 \log \left (1+\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{3 f^{3/2}}-\frac {2}{27} b^3 n^3 x^3 \log \left (d \left (e+f x^2\right )^m\right )+\frac {2}{9} b^2 n^2 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} b n 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} x^3 \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right )-\frac {2 b^2 (-e)^{3/2} m n^2 \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 {b (-e)^{3/2} m n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{f^{3/2}}+\frac {2 b^2 (-e)^{3/2} m n^2 \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 {b (-e)^{3/2} m n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{f^{3/2}}+\frac {2 i b^3 e^{3/2} m n^3 \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{9 f^{3/2}}-\frac {2 i b^3 e^{3/2} m n^3 \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{9 f^{3/2}}+\frac {2 b^3 (-e)^{3/2} m n^3 \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 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{f^{3/2}}-\frac {2 b^3 (-e)^{3/2} m n^3 \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 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{f^{3/2}}+\frac {2 b^3 (-e)^{3/2} m n^3 \text {Li}_4\left (-\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{f^{3/2}}-\frac {2 b^3 (-e)^{3/2} m n^3 \text {Li}_4\left (\frac {\sqrt {f} x}{\sqrt {-e}}\right )}{f^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(2544\) vs. \(2(1092)=2184\).
time = 0.57, size = 2544, normalized size = 2.33 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int x^{2} \left (a +b \ln \left (c \,x^{n}\right )\right )^{3} \ln \left (d \left (f \,x^{2}+e \right )^{m}\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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 )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________