Optimal. Leaf size=234 \[ -\frac {3 b f m n}{4 e x}-\frac {b f^2 m n \log (x)}{4 e^2}+\frac {b f^2 m n \log ^2(x)}{4 e^2}-\frac {f m \left (a+b \log \left (c x^n\right )\right )}{2 e x}-\frac {f^2 m \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac {b f^2 m n \log (e+f x)}{4 e^2}-\frac {b f^2 m n \log \left (-\frac {f x}{e}\right ) \log (e+f x)}{2 e^2}+\frac {f^2 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{2 e^2}-\frac {b n \log \left (d (e+f x)^m\right )}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{2 x^2}-\frac {b f^2 m n \text {Li}_2\left (1+\frac {f x}{e}\right )}{2 e^2} \]
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Rubi [A]
time = 0.11, antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2442, 46, 2423,
2338, 2441, 2352} \begin {gather*} -\frac {b f^2 m n \text {PolyLog}\left (2,\frac {f x}{e}+1\right )}{2 e^2}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{2 x^2}-\frac {f^2 m \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac {f^2 m \log (e+f x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac {f m \left (a+b \log \left (c x^n\right )\right )}{2 e x}-\frac {b n \log \left (d (e+f x)^m\right )}{4 x^2}+\frac {b f^2 m n \log ^2(x)}{4 e^2}-\frac {b f^2 m n \log (x)}{4 e^2}+\frac {b f^2 m n \log (e+f x)}{4 e^2}-\frac {b f^2 m n \log \left (-\frac {f x}{e}\right ) \log (e+f x)}{2 e^2}-\frac {3 b f m n}{4 e x} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 2338
Rule 2352
Rule 2423
Rule 2441
Rule 2442
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{x^3} \, dx &=-\frac {f m \left (a+b \log \left (c x^n\right )\right )}{2 e x}-\frac {f^2 m \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac {f^2 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{2 e^2}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{2 x^2}-(b n) \int \left (-\frac {f m}{2 e x^2}-\frac {f^2 m \log (x)}{2 e^2 x}+\frac {f^2 m \log (e+f x)}{2 e^2 x}-\frac {\log \left (d (e+f x)^m\right )}{2 x^3}\right ) \, dx\\ &=-\frac {b f m n}{2 e x}-\frac {f m \left (a+b \log \left (c x^n\right )\right )}{2 e x}-\frac {f^2 m \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac {f^2 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{2 e^2}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{2 x^2}+\frac {1}{2} (b n) \int \frac {\log \left (d (e+f x)^m\right )}{x^3} \, dx+\frac {\left (b f^2 m n\right ) \int \frac {\log (x)}{x} \, dx}{2 e^2}-\frac {\left (b f^2 m n\right ) \int \frac {\log (e+f x)}{x} \, dx}{2 e^2}\\ &=-\frac {b f m n}{2 e x}+\frac {b f^2 m n \log ^2(x)}{4 e^2}-\frac {f m \left (a+b \log \left (c x^n\right )\right )}{2 e x}-\frac {f^2 m \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac {b f^2 m n \log \left (-\frac {f x}{e}\right ) \log (e+f x)}{2 e^2}+\frac {f^2 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{2 e^2}-\frac {b n \log \left (d (e+f x)^m\right )}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{2 x^2}+\frac {1}{4} (b f m n) \int \frac {1}{x^2 (e+f x)} \, dx+\frac {\left (b f^3 m n\right ) \int \frac {\log \left (-\frac {f x}{e}\right )}{e+f x} \, dx}{2 e^2}\\ &=-\frac {b f m n}{2 e x}+\frac {b f^2 m n \log ^2(x)}{4 e^2}-\frac {f m \left (a+b \log \left (c x^n\right )\right )}{2 e x}-\frac {f^2 m \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}-\frac {b f^2 m n \log \left (-\frac {f x}{e}\right ) \log (e+f x)}{2 e^2}+\frac {f^2 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{2 e^2}-\frac {b n \log \left (d (e+f x)^m\right )}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{2 x^2}-\frac {b f^2 m n \text {Li}_2\left (1+\frac {f x}{e}\right )}{2 e^2}+\frac {1}{4} (b f m n) \int \left (\frac {1}{e x^2}-\frac {f}{e^2 x}+\frac {f^2}{e^2 (e+f x)}\right ) \, dx\\ &=-\frac {3 b f m n}{4 e x}-\frac {b f^2 m n \log (x)}{4 e^2}+\frac {b f^2 m n \log ^2(x)}{4 e^2}-\frac {f m \left (a+b \log \left (c x^n\right )\right )}{2 e x}-\frac {f^2 m \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac {b f^2 m n \log (e+f x)}{4 e^2}-\frac {b f^2 m n \log \left (-\frac {f x}{e}\right ) \log (e+f x)}{2 e^2}+\frac {f^2 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{2 e^2}-\frac {b n \log \left (d (e+f x)^m\right )}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{2 x^2}-\frac {b f^2 m n \text {Li}_2\left (1+\frac {f x}{e}\right )}{2 e^2}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 232, normalized size = 0.99 \begin {gather*} -\frac {2 a e f m x+3 b e f m n x-b f^2 m n x^2 \log ^2(x)+2 b e f m x \log \left (c x^n\right )-2 a f^2 m x^2 \log (e+f x)-b f^2 m n x^2 \log (e+f x)-2 b f^2 m x^2 \log \left (c x^n\right ) \log (e+f x)+2 a e^2 \log \left (d (e+f x)^m\right )+b e^2 n \log \left (d (e+f x)^m\right )+2 b e^2 \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+f^2 m x^2 \log (x) \left (2 a+b n+2 b \log \left (c x^n\right )+2 b n \log (e+f x)-2 b n \log \left (1+\frac {f x}{e}\right )\right )-2 b f^2 m n x^2 \text {Li}_2\left (-\frac {f x}{e}\right )}{4 e^2 x^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 = 0.32, size = 2100, normalized size = 8.97
method | result | size |
risch | \(\text {Expression too large to display}\) | \(2100\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.40, size = 271, normalized size = 1.16 \begin {gather*} \frac {1}{2} \, {\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 f^{2} m n e^{\left (-2\right )} + \frac {1}{4} \, {\left (2 \, a f^{2} m + {\left (f^{2} m n + 2 \, f^{2} m \log \left (c\right )\right )} b\right )} e^{\left (-2\right )} \log \left (f x + e\right ) - \frac {{\left (2 \, b f^{2} m n x^{2} \log \left (f x + e\right ) \log \left (x\right ) - b f^{2} m n x^{2} \log \left (x\right )^{2} + {\left (2 \, a f^{2} m + {\left (f^{2} m n + 2 \, f^{2} m \log \left (c\right )\right )} b\right )} x^{2} \log \left (x\right ) + {\left (2 \, a f m + {\left (3 \, f m n + 2 \, f m \log \left (c\right )\right )} b\right )} x e + {\left ({\left (n \log \left (d\right ) + 2 \, \log \left (c\right ) \log \left (d\right )\right )} b + 2 \, a \log \left (d\right )\right )} e^{2} + {\left (2 \, b e^{2} \log \left (x^{n}\right ) + {\left (b {\left (n + 2 \, \log \left (c\right )\right )} + 2 \, a\right )} e^{2}\right )} \log \left ({\left (f x + e\right )}^{m}\right ) - 2 \, {\left (b f^{2} m x^{2} \log \left (f x + e\right ) - b f^{2} m x^{2} \log \left (x\right ) - b f m x e - b e^{2} \log \left (d\right )\right )} \log \left (x^{n}\right )\right )} e^{\left (-2\right )}}{4 \, x^{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 \frac {\ln \left (d\,{\left (e+f\,x\right )}^m\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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