Optimal. Leaf size=274 \[ -\frac {5 b f m n}{36 e x^2}+\frac {4 b f^2 m n}{9 e^2 x}+\frac {b f^3 m n \log (x)}{9 e^3}-\frac {b f^3 m n \log ^2(x)}{6 e^3}-\frac {f m \left (a+b \log \left (c x^n\right )\right )}{6 e x^2}+\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )}{3 e^2 x}+\frac {f^3 m \log (x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac {b f^3 m n \log (e+f x)}{9 e^3}+\frac {b f^3 m n \log \left (-\frac {f x}{e}\right ) \log (e+f x)}{3 e^3}-\frac {f^3 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{3 e^3}-\frac {b n \log \left (d (e+f x)^m\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{3 x^3}+\frac {b f^3 m n \text {Li}_2\left (1+\frac {f x}{e}\right )}{3 e^3} \]
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Rubi [A]
time = 0.13, antiderivative size = 274, 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^3 m n \text {PolyLog}\left (2,\frac {f x}{e}+1\right )}{3 e^3}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{3 x^3}+\frac {f^3 m \log (x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac {f^3 m \log (e+f x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )}{3 e^2 x}-\frac {f m \left (a+b \log \left (c x^n\right )\right )}{6 e x^2}-\frac {b n \log \left (d (e+f x)^m\right )}{9 x^3}-\frac {b f^3 m n \log ^2(x)}{6 e^3}+\frac {b f^3 m n \log (x)}{9 e^3}-\frac {b f^3 m n \log (e+f x)}{9 e^3}+\frac {b f^3 m n \log \left (-\frac {f x}{e}\right ) \log (e+f x)}{3 e^3}+\frac {4 b f^2 m n}{9 e^2 x}-\frac {5 b f m n}{36 e x^2} \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^4} \, dx &=-\frac {f m \left (a+b \log \left (c x^n\right )\right )}{6 e x^2}+\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )}{3 e^2 x}+\frac {f^3 m \log (x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac {f^3 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{3 e^3}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{3 x^3}-(b n) \int \left (-\frac {f m}{6 e x^3}+\frac {f^2 m}{3 e^2 x^2}+\frac {f^3 m \log (x)}{3 e^3 x}-\frac {f^3 m \log (e+f x)}{3 e^3 x}-\frac {\log \left (d (e+f x)^m\right )}{3 x^4}\right ) \, dx\\ &=-\frac {b f m n}{12 e x^2}+\frac {b f^2 m n}{3 e^2 x}-\frac {f m \left (a+b \log \left (c x^n\right )\right )}{6 e x^2}+\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )}{3 e^2 x}+\frac {f^3 m \log (x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac {f^3 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{3 e^3}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{3 x^3}+\frac {1}{3} (b n) \int \frac {\log \left (d (e+f x)^m\right )}{x^4} \, dx-\frac {\left (b f^3 m n\right ) \int \frac {\log (x)}{x} \, dx}{3 e^3}+\frac {\left (b f^3 m n\right ) \int \frac {\log (e+f x)}{x} \, dx}{3 e^3}\\ &=-\frac {b f m n}{12 e x^2}+\frac {b f^2 m n}{3 e^2 x}-\frac {b f^3 m n \log ^2(x)}{6 e^3}-\frac {f m \left (a+b \log \left (c x^n\right )\right )}{6 e x^2}+\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )}{3 e^2 x}+\frac {f^3 m \log (x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {b f^3 m n \log \left (-\frac {f x}{e}\right ) \log (e+f x)}{3 e^3}-\frac {f^3 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{3 e^3}-\frac {b n \log \left (d (e+f x)^m\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{3 x^3}+\frac {1}{9} (b f m n) \int \frac {1}{x^3 (e+f x)} \, dx-\frac {\left (b f^4 m n\right ) \int \frac {\log \left (-\frac {f x}{e}\right )}{e+f x} \, dx}{3 e^3}\\ &=-\frac {b f m n}{12 e x^2}+\frac {b f^2 m n}{3 e^2 x}-\frac {b f^3 m n \log ^2(x)}{6 e^3}-\frac {f m \left (a+b \log \left (c x^n\right )\right )}{6 e x^2}+\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )}{3 e^2 x}+\frac {f^3 m \log (x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {b f^3 m n \log \left (-\frac {f x}{e}\right ) \log (e+f x)}{3 e^3}-\frac {f^3 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{3 e^3}-\frac {b n \log \left (d (e+f x)^m\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{3 x^3}+\frac {b f^3 m n \text {Li}_2\left (1+\frac {f x}{e}\right )}{3 e^3}+\frac {1}{9} (b f m n) \int \left (\frac {1}{e x^3}-\frac {f}{e^2 x^2}+\frac {f^2}{e^3 x}-\frac {f^3}{e^3 (e+f x)}\right ) \, dx\\ &=-\frac {5 b f m n}{36 e x^2}+\frac {4 b f^2 m n}{9 e^2 x}+\frac {b f^3 m n \log (x)}{9 e^3}-\frac {b f^3 m n \log ^2(x)}{6 e^3}-\frac {f m \left (a+b \log \left (c x^n\right )\right )}{6 e x^2}+\frac {f^2 m \left (a+b \log \left (c x^n\right )\right )}{3 e^2 x}+\frac {f^3 m \log (x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac {b f^3 m n \log (e+f x)}{9 e^3}+\frac {b f^3 m n \log \left (-\frac {f x}{e}\right ) \log (e+f x)}{3 e^3}-\frac {f^3 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{3 e^3}-\frac {b n \log \left (d (e+f x)^m\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{3 x^3}+\frac {b f^3 m n \text {Li}_2\left (1+\frac {f x}{e}\right )}{3 e^3}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 280, normalized size = 1.02 \begin {gather*} -\frac {6 a e^2 f m x+5 b e^2 f m n x-12 a e f^2 m x^2-16 b e f^2 m n x^2+6 b f^3 m n x^3 \log ^2(x)+6 b e^2 f m x \log \left (c x^n\right )-12 b e f^2 m x^2 \log \left (c x^n\right )+12 a f^3 m x^3 \log (e+f x)+4 b f^3 m n x^3 \log (e+f x)+12 b f^3 m x^3 \log \left (c x^n\right ) \log (e+f x)+12 a e^3 \log \left (d (e+f x)^m\right )+4 b e^3 n \log \left (d (e+f x)^m\right )+12 b e^3 \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )-4 f^3 m x^3 \log (x) \left (3 a+b n+3 b \log \left (c x^n\right )+3 b n \log (e+f x)-3 b n \log \left (1+\frac {f x}{e}\right )\right )+12 b f^3 m n x^3 \text {Li}_2\left (-\frac {f x}{e}\right )}{36 e^3 x^3} \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 = 0.36, size = 2282, normalized size = 8.33
method | result | size |
risch | \(\text {Expression too large to display}\) | \(2282\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.40, size = 320, normalized size = 1.17 \begin {gather*} -\frac {1}{3} \, {\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^{3} m n e^{\left (-3\right )} - \frac {1}{9} \, {\left (3 \, a f^{3} m + {\left (f^{3} m n + 3 \, f^{3} m \log \left (c\right )\right )} b\right )} e^{\left (-3\right )} \log \left (f x + e\right ) + \frac {{\left (12 \, b f^{3} m n x^{3} \log \left (f x + e\right ) \log \left (x\right ) - 6 \, b f^{3} m n x^{3} \log \left (x\right )^{2} + 4 \, {\left (3 \, a f^{3} m + {\left (f^{3} m n + 3 \, f^{3} m \log \left (c\right )\right )} b\right )} x^{3} \log \left (x\right ) + 4 \, {\left (3 \, a f^{2} m + {\left (4 \, f^{2} m n + 3 \, f^{2} m \log \left (c\right )\right )} b\right )} x^{2} e - {\left (6 \, a f m + {\left (5 \, f m n + 6 \, f m \log \left (c\right )\right )} b\right )} x e^{2} - 4 \, {\left ({\left (n \log \left (d\right ) + 3 \, \log \left (c\right ) \log \left (d\right )\right )} b + 3 \, a \log \left (d\right )\right )} e^{3} - 4 \, {\left (3 \, b e^{3} \log \left (x^{n}\right ) + {\left (b {\left (n + 3 \, \log \left (c\right )\right )} + 3 \, a\right )} e^{3}\right )} \log \left ({\left (f x + e\right )}^{m}\right ) - 6 \, {\left (2 \, b f^{3} m x^{3} \log \left (f x + e\right ) - 2 \, b f^{3} m x^{3} \log \left (x\right ) - 2 \, b f^{2} m x^{2} e + b f m x e^{2} + 2 \, b e^{3} \log \left (d\right )\right )} \log \left (x^{n}\right )\right )} e^{\left (-3\right )}}{36 \, x^{3}} \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^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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