Optimal. Leaf size=304 \[ \frac {b f k n x^m (g x)^{-m} \log (x)}{e g m}-\frac {b f k n x^m (g x)^{-m} \log ^2(x)}{2 e g}+\frac {f k x^m (g x)^{-m} \log (x) \left (a+b \log \left (c x^n\right )\right )}{e g}-\frac {b f k n x^m (g x)^{-m} \log \left (e+f x^m\right )}{e g m^2}+\frac {b f k n x^m (g x)^{-m} \log \left (-\frac {f x^m}{e}\right ) \log \left (e+f x^m\right )}{e g m^2}-\frac {f k x^m (g x)^{-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{e g m}-\frac {b n (g x)^{-m} \log \left (d \left (e+f x^m\right )^k\right )}{g m^2}-\frac {(g x)^{-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{g m}+\frac {b f k n x^m (g x)^{-m} \text {Li}_2\left (1+\frac {f x^m}{e}\right )}{e g m^2} \]
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Rubi [A]
time = 0.21, antiderivative size = 304, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 12, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2505, 19,
272, 36, 29, 31, 2423, 2338, 2504, 2441, 2352, 16} \begin {gather*} \frac {b f k n x^m (g x)^{-m} \text {PolyLog}\left (2,\frac {f x^m}{e}+1\right )}{e g m^2}-\frac {(g x)^{-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{g m}+\frac {f k x^m \log (x) (g x)^{-m} \left (a+b \log \left (c x^n\right )\right )}{e g}-\frac {f k x^m (g x)^{-m} \log \left (e+f x^m\right ) \left (a+b \log \left (c x^n\right )\right )}{e g m}-\frac {b n (g x)^{-m} \log \left (d \left (e+f x^m\right )^k\right )}{g m^2}-\frac {b f k n x^m (g x)^{-m} \log \left (e+f x^m\right )}{e g m^2}+\frac {b f k n x^m (g x)^{-m} \log \left (-\frac {f x^m}{e}\right ) \log \left (e+f x^m\right )}{e g m^2}-\frac {b f k n x^m \log ^2(x) (g x)^{-m}}{2 e g}+\frac {b f k n x^m \log (x) (g x)^{-m}}{e g m} \end {gather*}
Antiderivative was successfully verified.
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Rule 16
Rule 19
Rule 29
Rule 31
Rule 36
Rule 272
Rule 2338
Rule 2352
Rule 2423
Rule 2441
Rule 2504
Rule 2505
Rubi steps
\begin {align*} \int (g x)^{-1-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ) \, dx &=\frac {f k x^m (g x)^{-m} \log (x) \left (a+b \log \left (c x^n\right )\right )}{e g}-\frac {f k x^m (g x)^{-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{e g m}-\frac {(g x)^{-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{g m}-(b n) \int \left (\frac {f k x^{-1+m} (g x)^{-m} \log (x)}{e g}-\frac {f k x^{-1+m} (g x)^{-m} \log \left (e+f x^m\right )}{e g m}-\frac {(g x)^{-m} \log \left (d \left (e+f x^m\right )^k\right )}{g m x}\right ) \, dx\\ &=\frac {f k x^m (g x)^{-m} \log (x) \left (a+b \log \left (c x^n\right )\right )}{e g}-\frac {f k x^m (g x)^{-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{e g m}-\frac {(g x)^{-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{g m}-\frac {(b f k n) \int x^{-1+m} (g x)^{-m} \log (x) \, dx}{e g}+\frac {(b n) \int \frac {(g x)^{-m} \log \left (d \left (e+f x^m\right )^k\right )}{x} \, dx}{g m}+\frac {(b f k n) \int x^{-1+m} (g x)^{-m} \log \left (e+f x^m\right ) \, dx}{e g m}\\ &=\frac {f k x^m (g x)^{-m} \log (x) \left (a+b \log \left (c x^n\right )\right )}{e g}-\frac {f k x^m (g x)^{-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{e g m}-\frac {(g x)^{-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{g m}+\frac {(b n) \int (g x)^{-1-m} \log \left (d \left (e+f x^m\right )^k\right ) \, dx}{m}-\frac {\left (b f k n x^m (g x)^{-m}\right ) \int \frac {\log (x)}{x} \, dx}{e g}+\frac {\left (b f k n x^m (g x)^{-m}\right ) \int \frac {\log \left (e+f x^m\right )}{x} \, dx}{e g m}\\ &=-\frac {b f k n x^m (g x)^{-m} \log ^2(x)}{2 e g}+\frac {f k x^m (g x)^{-m} \log (x) \left (a+b \log \left (c x^n\right )\right )}{e g}-\frac {f k x^m (g x)^{-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{e g m}-\frac {b n (g x)^{-m} \log \left (d \left (e+f x^m\right )^k\right )}{g m^2}-\frac {(g x)^{-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{g m}+\frac {(b f k n) \int \frac {x^{-1+m} (g x)^{-m}}{e+f x^m} \, dx}{g m}+\frac {\left (b f k n x^m (g x)^{-m}\right ) \text {Subst}\left (\int \frac {\log (e+f x)}{x} \, dx,x,x^m\right )}{e g m^2}\\ &=-\frac {b f k n x^m (g x)^{-m} \log ^2(x)}{2 e g}+\frac {f k x^m (g x)^{-m} \log (x) \left (a+b \log \left (c x^n\right )\right )}{e g}+\frac {b f k n x^m (g x)^{-m} \log \left (-\frac {f x^m}{e}\right ) \log \left (e+f x^m\right )}{e g m^2}-\frac {f k x^m (g x)^{-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{e g m}-\frac {b n (g x)^{-m} \log \left (d \left (e+f x^m\right )^k\right )}{g m^2}-\frac {(g x)^{-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{g m}-\frac {\left (b f^2 k n x^m (g x)^{-m}\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {f x}{e}\right )}{e+f x} \, dx,x,x^m\right )}{e g m^2}+\frac {\left (b f k n x^m (g x)^{-m}\right ) \int \frac {1}{x \left (e+f x^m\right )} \, dx}{g m}\\ &=-\frac {b f k n x^m (g x)^{-m} \log ^2(x)}{2 e g}+\frac {f k x^m (g x)^{-m} \log (x) \left (a+b \log \left (c x^n\right )\right )}{e g}+\frac {b f k n x^m (g x)^{-m} \log \left (-\frac {f x^m}{e}\right ) \log \left (e+f x^m\right )}{e g m^2}-\frac {f k x^m (g x)^{-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{e g m}-\frac {b n (g x)^{-m} \log \left (d \left (e+f x^m\right )^k\right )}{g m^2}-\frac {(g x)^{-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{g m}+\frac {b f k n x^m (g x)^{-m} \text {Li}_2\left (1+\frac {f x^m}{e}\right )}{e g m^2}+\frac {\left (b f k n x^m (g x)^{-m}\right ) \text {Subst}\left (\int \frac {1}{x (e+f x)} \, dx,x,x^m\right )}{g m^2}\\ &=-\frac {b f k n x^m (g x)^{-m} \log ^2(x)}{2 e g}+\frac {f k x^m (g x)^{-m} \log (x) \left (a+b \log \left (c x^n\right )\right )}{e g}+\frac {b f k n x^m (g x)^{-m} \log \left (-\frac {f x^m}{e}\right ) \log \left (e+f x^m\right )}{e g m^2}-\frac {f k x^m (g x)^{-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{e g m}-\frac {b n (g x)^{-m} \log \left (d \left (e+f x^m\right )^k\right )}{g m^2}-\frac {(g x)^{-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{g m}+\frac {b f k n x^m (g x)^{-m} \text {Li}_2\left (1+\frac {f x^m}{e}\right )}{e g m^2}+\frac {\left (b f k n x^m (g x)^{-m}\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^m\right )}{e g m^2}-\frac {\left (b f^2 k n x^m (g x)^{-m}\right ) \text {Subst}\left (\int \frac {1}{e+f x} \, dx,x,x^m\right )}{e g m^2}\\ &=\frac {b f k n x^m (g x)^{-m} \log (x)}{e g m}-\frac {b f k n x^m (g x)^{-m} \log ^2(x)}{2 e g}+\frac {f k x^m (g x)^{-m} \log (x) \left (a+b \log \left (c x^n\right )\right )}{e g}-\frac {b f k n x^m (g x)^{-m} \log \left (e+f x^m\right )}{e g m^2}+\frac {b f k n x^m (g x)^{-m} \log \left (-\frac {f x^m}{e}\right ) \log \left (e+f x^m\right )}{e g m^2}-\frac {f k x^m (g x)^{-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{e g m}-\frac {b n (g x)^{-m} \log \left (d \left (e+f x^m\right )^k\right )}{g m^2}-\frac {(g x)^{-m} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{g m}+\frac {b f k n x^m (g x)^{-m} \text {Li}_2\left (1+\frac {f x^m}{e}\right )}{e g m^2}\\ \end {align*}
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Mathematica [A]
time = 0.22, size = 162, normalized size = 0.53 \begin {gather*} \frac {(g x)^{-m} \left (-b f k m^2 n x^m \log ^2(x)-2 \left (a m+b n+b m \log \left (c x^n\right )\right ) \left (f k x^m \log \left (f-f x^{-m}\right )+e \log \left (d \left (e+f x^m\right )^k\right )\right )+2 f k m x^m \log (x) \left (a m+b n+b m \log \left (c x^n\right )+b n \log \left (f-f x^{-m}\right )-b n \log \left (1+\frac {f x^m}{e}\right )\right )-2 b f k n x^m \text {Li}_2\left (-\frac {f x^m}{e}\right )\right )}{2 e g m^2} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.10, size = 0, normalized size = 0.00 \[\int \left (g x \right )^{-m -1} \left (a +b \ln \left (c \,x^{n}\right )\right ) \ln \left (d \left (e +f \,x^{m}\right )^{k}\right )\, dx\]
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 [A]
time = 0.38, size = 247, normalized size = 0.81 \begin {gather*} -\frac {{\left (2 \, b f g^{-m - 1} k m n x^{m} \log \left ({\left (f x^{m} + e\right )} e^{\left (-1\right )}\right ) \log \left (x\right ) + 2 \, b f g^{-m - 1} k n x^{m} {\rm Li}_2\left (-{\left (f x^{m} + e\right )} e^{\left (-1\right )} + 1\right ) - {\left (b f k m^{2} n \log \left (x\right )^{2} + 2 \, {\left (b f k m^{2} \log \left (c\right ) + a f k m^{2} + b f k m n\right )} \log \left (x\right )\right )} g^{-m - 1} x^{m} + 2 \, {\left (b m n e \log \left (d\right ) \log \left (x\right ) + {\left (b m e \log \left (c\right ) + {\left (a m + b n\right )} e\right )} \log \left (d\right )\right )} g^{-m - 1} + 2 \, {\left ({\left (b f k m \log \left (c\right ) + a f k m + b f k n\right )} g^{-m - 1} x^{m} + {\left (b k m n e \log \left (x\right ) + b k m e \log \left (c\right ) + {\left (a k m + b k n\right )} e\right )} g^{-m - 1}\right )} \log \left (f x^{m} + e\right )\right )} e^{\left (-1\right )}}{2 \, m^{2} x^{m}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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^m\right )}^k\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (g\,x\right )}^{m+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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