Optimal. Leaf size=221 \[ -\frac {3 b e m n x^2}{16 f}+\frac {1}{16} b m n x^4+\frac {e m x^2 \left (a+b \log \left (c x^n\right )\right )}{4 f}-\frac {1}{8} m x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {b e^2 m n \log \left (e+f x^2\right )}{16 f^2}+\frac {b e^2 m n \log \left (-\frac {f x^2}{e}\right ) \log \left (e+f x^2\right )}{8 f^2}-\frac {e^2 m \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^2\right )}{4 f^2}-\frac {1}{16} b n x^4 \log \left (d \left (e+f x^2\right )^m\right )+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac {b e^2 m n \text {Li}_2\left (1+\frac {f x^2}{e}\right )}{8 f^2} \]
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Rubi [A]
time = 0.16, antiderivative size = 221, 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 = {2504, 2442, 45,
2423, 2441, 2352} \begin {gather*} \frac {b e^2 m n \text {PolyLog}\left (2,\frac {f x^2}{e}+1\right )}{8 f^2}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac {e^2 m \log \left (e+f x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{4 f^2}+\frac {e m x^2 \left (a+b \log \left (c x^n\right )\right )}{4 f}-\frac {1}{8} m x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{16} b n x^4 \log \left (d \left (e+f x^2\right )^m\right )+\frac {b e^2 m n \log \left (e+f x^2\right )}{16 f^2}+\frac {b e^2 m n \log \left (-\frac {f x^2}{e}\right ) \log \left (e+f x^2\right )}{8 f^2}-\frac {3 b e m n x^2}{16 f}+\frac {1}{16} b m n x^4 \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2352
Rule 2423
Rule 2441
Rule 2442
Rule 2504
Rubi steps
\begin {align*} \int x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right ) \, dx &=\frac {e m x^2 \left (a+b \log \left (c x^n\right )\right )}{4 f}-\frac {1}{8} m x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {e^2 m \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^2\right )}{4 f^2}+\frac {1}{4} x^4 \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 {e m x}{4 f}-\frac {m x^3}{8}-\frac {e^2 m \log \left (e+f x^2\right )}{4 f^2 x}+\frac {1}{4} x^3 \log \left (d \left (e+f x^2\right )^m\right )\right ) \, dx\\ &=-\frac {b e m n x^2}{8 f}+\frac {1}{32} b m n x^4+\frac {e m x^2 \left (a+b \log \left (c x^n\right )\right )}{4 f}-\frac {1}{8} m x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {e^2 m \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^2\right )}{4 f^2}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac {1}{4} (b n) \int x^3 \log \left (d \left (e+f x^2\right )^m\right ) \, dx+\frac {\left (b e^2 m n\right ) \int \frac {\log \left (e+f x^2\right )}{x} \, dx}{4 f^2}\\ &=-\frac {b e m n x^2}{8 f}+\frac {1}{32} b m n x^4+\frac {e m x^2 \left (a+b \log \left (c x^n\right )\right )}{4 f}-\frac {1}{8} m x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {e^2 m \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^2\right )}{4 f^2}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac {1}{8} (b n) \text {Subst}\left (\int x \log \left (d (e+f x)^m\right ) \, dx,x,x^2\right )+\frac {\left (b e^2 m n\right ) \text {Subst}\left (\int \frac {\log (e+f x)}{x} \, dx,x,x^2\right )}{8 f^2}\\ &=-\frac {b e m n x^2}{8 f}+\frac {1}{32} b m n x^4+\frac {e m x^2 \left (a+b \log \left (c x^n\right )\right )}{4 f}-\frac {1}{8} m x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {b e^2 m n \log \left (-\frac {f x^2}{e}\right ) \log \left (e+f x^2\right )}{8 f^2}-\frac {e^2 m \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^2\right )}{4 f^2}-\frac {1}{16} b n x^4 \log \left (d \left (e+f x^2\right )^m\right )+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac {\left (b e^2 m n\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {f x}{e}\right )}{e+f x} \, dx,x,x^2\right )}{8 f}+\frac {1}{16} (b f m n) \text {Subst}\left (\int \frac {x^2}{e+f x} \, dx,x,x^2\right )\\ &=-\frac {b e m n x^2}{8 f}+\frac {1}{32} b m n x^4+\frac {e m x^2 \left (a+b \log \left (c x^n\right )\right )}{4 f}-\frac {1}{8} m x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {b e^2 m n \log \left (-\frac {f x^2}{e}\right ) \log \left (e+f x^2\right )}{8 f^2}-\frac {e^2 m \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^2\right )}{4 f^2}-\frac {1}{16} b n x^4 \log \left (d \left (e+f x^2\right )^m\right )+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac {b e^2 m n \text {Li}_2\left (1+\frac {f x^2}{e}\right )}{8 f^2}+\frac {1}{16} (b f m n) \text {Subst}\left (\int \left (-\frac {e}{f^2}+\frac {x}{f}+\frac {e^2}{f^2 (e+f x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {3 b e m n x^2}{16 f}+\frac {1}{16} b m n x^4+\frac {e m x^2 \left (a+b \log \left (c x^n\right )\right )}{4 f}-\frac {1}{8} m x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {b e^2 m n \log \left (e+f x^2\right )}{16 f^2}+\frac {b e^2 m n \log \left (-\frac {f x^2}{e}\right ) \log \left (e+f x^2\right )}{8 f^2}-\frac {e^2 m \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^2\right )}{4 f^2}-\frac {1}{16} b n x^4 \log \left (d \left (e+f x^2\right )^m\right )+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )+\frac {b e^2 m n \text {Li}_2\left (1+\frac {f x^2}{e}\right )}{8 f^2}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.11, size = 324, normalized size = 1.47 \begin {gather*} -\frac {-4 a e f m x^2+3 b e f m n x^2+2 a f^2 m x^4-b f^2 m n x^4-4 b e f m x^2 \log \left (c x^n\right )+2 b f^2 m x^4 \log \left (c x^n\right )+4 b e^2 m n \log (x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+4 b e^2 m n \log (x) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )+4 a e^2 m \log \left (e+f x^2\right )-b e^2 m n \log \left (e+f x^2\right )-4 b e^2 m n \log (x) \log \left (e+f x^2\right )+4 b e^2 m \log \left (c x^n\right ) \log \left (e+f x^2\right )-4 a f^2 x^4 \log \left (d \left (e+f x^2\right )^m\right )+b f^2 n x^4 \log \left (d \left (e+f x^2\right )^m\right )-4 b f^2 x^4 \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+4 b e^2 m n \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+4 b e^2 m n \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{16 f^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 = 0.69, size = 2270, normalized size = 10.27
method | result | size |
risch | \(\text {Expression too large to display}\) | \(2270\) |
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^3\,\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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