Optimal. Leaf size=251 \[ -\frac {8 b e m n x}{9 f}+\frac {4}{27} b m n x^3+\frac {2 b e^{3/2} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{9 f^{3/2}}+\frac {2 e m x \left (a+b \log \left (c x^n\right )\right )}{3 f}-\frac {2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {2 e^{3/2} m \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^{3/2}}-\frac {1}{9} b n x^3 \log \left (d \left (e+f x^2\right )^m\right )+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac {i b e^{3/2} m n \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{3 f^{3/2}}-\frac {i b e^{3/2} m n \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{3 f^{3/2}} \]
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Rubi [A]
time = 0.13, antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2505, 308, 211,
2423, 4940, 2438} \begin {gather*} \frac {i b e^{3/2} m n \text {PolyLog}\left (2,-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{3 f^{3/2}}-\frac {i b e^{3/2} m n \text {PolyLog}\left (2,\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{3 f^{3/2}}-\frac {2 e^{3/2} m \text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^{3/2}}+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac {2 e m x \left (a+b \log \left (c x^n\right )\right )}{3 f}-\frac {2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {2 b e^{3/2} m n \text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{9 f^{3/2}}-\frac {1}{9} b n x^3 \log \left (d \left (e+f x^2\right )^m\right )-\frac {8 b e m n x}{9 f}+\frac {4}{27} b m n x^3 \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 308
Rule 2423
Rule 2438
Rule 2505
Rule 4940
Rubi steps
\begin {align*} \int x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right ) \, dx &=\frac {2 e m x \left (a+b \log \left (c x^n\right )\right )}{3 f}-\frac {2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {2 e^{3/2} m \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^{3/2}}+\frac {1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-(b n) \int \left (\frac {2 e m}{3 f}-\frac {2 m x^2}{9}-\frac {2 e^{3/2} m \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{3 f^{3/2} x}+\frac {1}{3} x^2 \log \left (d \left (e+f x^2\right )^m\right )\right ) \, dx\\ &=-\frac {2 b e m n x}{3 f}+\frac {2}{27} b m n x^3+\frac {2 e m x \left (a+b \log \left (c x^n\right )\right )}{3 f}-\frac {2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {2 e^{3/2} m \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^{3/2}}+\frac {1}{3} 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) \int x^2 \log \left (d \left (e+f x^2\right )^m\right ) \, dx+\frac {\left (2 b e^{3/2} m n\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{x} \, dx}{3 f^{3/2}}\\ &=-\frac {2 b e m n x}{3 f}+\frac {2}{27} b m n x^3+\frac {2 e m x \left (a+b \log \left (c x^n\right )\right )}{3 f}-\frac {2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {2 e^{3/2} m \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^{3/2}}-\frac {1}{9} b n x^3 \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 ) \log \left (d \left (e+f x^2\right )^m\right )+\frac {\left (i b e^{3/2} m n\right ) \int \frac {\log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{x} \, dx}{3 f^{3/2}}-\frac {\left (i b e^{3/2} m n\right ) \int \frac {\log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{x} \, dx}{3 f^{3/2}}+\frac {1}{9} (2 b f m n) \int \frac {x^4}{e+f x^2} \, dx\\ &=-\frac {2 b e m n x}{3 f}+\frac {2}{27} b m n x^3+\frac {2 e m x \left (a+b \log \left (c x^n\right )\right )}{3 f}-\frac {2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {2 e^{3/2} m \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^{3/2}}-\frac {1}{9} b n x^3 \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 ) \log \left (d \left (e+f x^2\right )^m\right )+\frac {i b e^{3/2} m n \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{3 f^{3/2}}-\frac {i b e^{3/2} m n \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{3 f^{3/2}}+\frac {1}{9} (2 b f m n) \int \left (-\frac {e}{f^2}+\frac {x^2}{f}+\frac {e^2}{f^2 \left (e+f x^2\right )}\right ) \, dx\\ &=-\frac {8 b e m n x}{9 f}+\frac {4}{27} b m n x^3+\frac {2 e m x \left (a+b \log \left (c x^n\right )\right )}{3 f}-\frac {2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {2 e^{3/2} m \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^{3/2}}-\frac {1}{9} b n x^3 \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 ) \log \left (d \left (e+f x^2\right )^m\right )+\frac {i b e^{3/2} m n \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{3 f^{3/2}}-\frac {i b e^{3/2} m n \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{3 f^{3/2}}+\frac {\left (2 b e^2 m n\right ) \int \frac {1}{e+f x^2} \, dx}{9 f}\\ &=-\frac {8 b e m n x}{9 f}+\frac {4}{27} b m n x^3+\frac {2 b e^{3/2} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{9 f^{3/2}}+\frac {2 e m x \left (a+b \log \left (c x^n\right )\right )}{3 f}-\frac {2}{9} m x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {2 e^{3/2} m \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^{3/2}}-\frac {1}{9} b n x^3 \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 ) \log \left (d \left (e+f x^2\right )^m\right )+\frac {i b e^{3/2} m n \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{3 f^{3/2}}-\frac {i b e^{3/2} m n \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{3 f^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 389, normalized size = 1.55 \begin {gather*} \frac {18 a e \sqrt {f} m x-24 b e \sqrt {f} m n x-6 a f^{3/2} m x^3+4 b f^{3/2} m n x^3-18 a e^{3/2} m \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )+6 b e^{3/2} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )+18 b e^{3/2} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log (x)+18 b e \sqrt {f} m x \log \left (c x^n\right )-6 b f^{3/2} m x^3 \log \left (c x^n\right )-18 b e^{3/2} m \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log \left (c x^n\right )-9 i b e^{3/2} m n \log (x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+9 i b e^{3/2} m n \log (x) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )+9 a f^{3/2} x^3 \log \left (d \left (e+f x^2\right )^m\right )-3 b f^{3/2} n x^3 \log \left (d \left (e+f x^2\right )^m\right )+9 b f^{3/2} x^3 \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+9 i b e^{3/2} m n \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-9 i b e^{3/2} m n \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{27 f^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.19, size = 2321, normalized size = 9.25
method | result | size |
risch | \(\text {Expression too large to display}\) | \(2321\) |
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 ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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