Optimal. Leaf size=203 \[ -\frac {3 b e m n x}{4 f}+\frac {1}{4} b m n x^2+\frac {e m x \left (a+b \log \left (c x^n\right )\right )}{2 f}-\frac {1}{4} m x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {b e^2 m n \log (e+f x)}{4 f^2}+\frac {b e^2 m n \log \left (-\frac {f x}{e}\right ) \log (e+f x)}{2 f^2}-\frac {e^2 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{2 f^2}-\frac {1}{4} b n x^2 \log \left (d (e+f x)^m\right )+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac {b e^2 m n \text {Li}_2\left (1+\frac {f x}{e}\right )}{2 f^2} \]
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Rubi [A]
time = 0.09, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {2442, 45, 2423,
2441, 2352} \begin {gather*} \frac {b e^2 m n \text {PolyLog}\left (2,\frac {f x}{e}+1\right )}{2 f^2}+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )-\frac {e^2 m \log (e+f x) \left (a+b \log \left (c x^n\right )\right )}{2 f^2}+\frac {e m x \left (a+b \log \left (c x^n\right )\right )}{2 f}-\frac {1}{4} m x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} b n x^2 \log \left (d (e+f x)^m\right )+\frac {b e^2 m n \log (e+f x)}{4 f^2}+\frac {b e^2 m n \log \left (-\frac {f x}{e}\right ) \log (e+f x)}{2 f^2}-\frac {3 b e m n x}{4 f}+\frac {1}{4} b m n x^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2352
Rule 2423
Rule 2441
Rule 2442
Rubi steps
\begin {align*} \int x \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right ) \, dx &=\frac {e m x \left (a+b \log \left (c x^n\right )\right )}{2 f}-\frac {1}{4} m x^2 \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 (e+f x)}{2 f^2}+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )-(b n) \int \left (\frac {e m}{2 f}-\frac {m x}{4}-\frac {e^2 m \log (e+f x)}{2 f^2 x}+\frac {1}{2} x \log \left (d (e+f x)^m\right )\right ) \, dx\\ &=-\frac {b e m n x}{2 f}+\frac {1}{8} b m n x^2+\frac {e m x \left (a+b \log \left (c x^n\right )\right )}{2 f}-\frac {1}{4} m x^2 \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 (e+f x)}{2 f^2}+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )-\frac {1}{2} (b n) \int x \log \left (d (e+f x)^m\right ) \, dx+\frac {\left (b e^2 m n\right ) \int \frac {\log (e+f x)}{x} \, dx}{2 f^2}\\ &=-\frac {b e m n x}{2 f}+\frac {1}{8} b m n x^2+\frac {e m x \left (a+b \log \left (c x^n\right )\right )}{2 f}-\frac {1}{4} m x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {b e^2 m n \log \left (-\frac {f x}{e}\right ) \log (e+f x)}{2 f^2}-\frac {e^2 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{2 f^2}-\frac {1}{4} b n x^2 \log \left (d (e+f x)^m\right )+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )-\frac {\left (b e^2 m n\right ) \int \frac {\log \left (-\frac {f x}{e}\right )}{e+f x} \, dx}{2 f}+\frac {1}{4} (b f m n) \int \frac {x^2}{e+f x} \, dx\\ &=-\frac {b e m n x}{2 f}+\frac {1}{8} b m n x^2+\frac {e m x \left (a+b \log \left (c x^n\right )\right )}{2 f}-\frac {1}{4} m x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {b e^2 m n \log \left (-\frac {f x}{e}\right ) \log (e+f x)}{2 f^2}-\frac {e^2 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{2 f^2}-\frac {1}{4} b n x^2 \log \left (d (e+f x)^m\right )+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac {b e^2 m n \text {Li}_2\left (1+\frac {f x}{e}\right )}{2 f^2}+\frac {1}{4} (b f m n) \int \left (-\frac {e}{f^2}+\frac {x}{f}+\frac {e^2}{f^2 (e+f x)}\right ) \, dx\\ &=-\frac {3 b e m n x}{4 f}+\frac {1}{4} b m n x^2+\frac {e m x \left (a+b \log \left (c x^n\right )\right )}{2 f}-\frac {1}{4} m x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {b e^2 m n \log (e+f x)}{4 f^2}+\frac {b e^2 m n \log \left (-\frac {f x}{e}\right ) \log (e+f x)}{2 f^2}-\frac {e^2 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{2 f^2}-\frac {1}{4} b n x^2 \log \left (d (e+f x)^m\right )+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac {b e^2 m n \text {Li}_2\left (1+\frac {f x}{e}\right )}{2 f^2}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 208, normalized size = 1.02 \begin {gather*} \frac {2 a e f m x-3 b e f m n x-a f^2 m x^2+b f^2 m n x^2-2 a e^2 m \log (e+f x)+b e^2 m n \log (e+f x)+2 b e^2 m n \log (x) \log (e+f x)+2 a f^2 x^2 \log \left (d (e+f x)^m\right )-b f^2 n x^2 \log \left (d (e+f x)^m\right )+b \log \left (c x^n\right ) \left (-2 e^2 m \log (e+f x)+f x \left (2 e m-f m x+2 f x \log \left (d (e+f x)^m\right )\right )\right )-2 b e^2 m n \log (x) \log \left (1+\frac {f x}{e}\right )-2 b e^2 m n \text {Li}_2\left (-\frac {f x}{e}\right )}{4 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 = 0.38, size = 2041, normalized size = 10.05
method | result | size |
risch | \(\text {Expression too large to display}\) | \(2041\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.39, size = 260, normalized size = 1.28 \begin {gather*} -\frac {{\left (\log \left (f x e^{\left (-1\right )} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-f x e^{\left (-1\right )}\right )\right )} b m n e^{2}}{2 \, f^{2}} + \frac {{\left ({\left (m n - 2 \, m \log \left (c\right )\right )} b - 2 \, a m\right )} e^{2} \log \left (f x + e\right )}{4 \, f^{2}} + \frac {2 \, b m n e^{2} \log \left (f x + e\right ) \log \left (x\right ) - {\left ({\left (f^{2} m - 2 \, f^{2} \log \left (d\right )\right )} a - {\left (f^{2} m n - f^{2} n \log \left (d\right ) - {\left (f^{2} m - 2 \, f^{2} \log \left (d\right )\right )} \log \left (c\right )\right )} b\right )} x^{2} + {\left (2 \, a f m - {\left (3 \, f m n - 2 \, f m \log \left (c\right )\right )} b\right )} x e + {\left (2 \, b f^{2} x^{2} \log \left (x^{n}\right ) + {\left (2 \, a f^{2} - {\left (f^{2} n - 2 \, f^{2} \log \left (c\right )\right )} b\right )} x^{2}\right )} \log \left ({\left (f x + e\right )}^{m}\right ) + {\left (2 \, b f m x e - {\left (f^{2} m - 2 \, f^{2} \log \left (d\right )\right )} b x^{2} - 2 \, b m e^{2} \log \left (f x + e\right )\right )} \log \left (x^{n}\right )}{4 \, f^{2}} \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\,\ln \left (d\,{\left (e+f\,x\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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