Optimal. Leaf size=255 \[ \frac {2 b k n (g x)^m}{g m^2}-\frac {k (g x)^m \left (a+b \log \left (c x^n\right )\right )}{g m}-\frac {b e k n x^{-m} (g x)^m \log \left (e+f x^m\right )}{f g m^2}-\frac {b e k n x^{-m} (g x)^m \log \left (-\frac {f x^m}{e}\right ) \log \left (e+f x^m\right )}{f g m^2}+\frac {e k x^{-m} (g x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{f g m}-\frac {b n (g x)^m \log \left (d \left (e+f x^m\right )^k\right )}{g m^2}+\frac {(g x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right )}{g m}-\frac {b e k n x^{-m} (g x)^m \text {Li}_2\left (1+\frac {f x^m}{e}\right )}{f g m^2} \]
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Rubi [A]
time = 0.16, antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 11, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {2505, 20,
272, 45, 2423, 16, 32, 19, 2504, 2441, 2352} \begin {gather*} -\frac {b e k n x^{-m} (g x)^m \text {PolyLog}\left (2,\frac {f x^m}{e}+1\right )}{f 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 {e k x^{-m} (g x)^m \log \left (e+f x^m\right ) \left (a+b \log \left (c x^n\right )\right )}{f g m}-\frac {k (g x)^m \left (a+b \log \left (c x^n\right )\right )}{g m}-\frac {b n (g x)^m \log \left (d \left (e+f x^m\right )^k\right )}{g m^2}-\frac {b e k n x^{-m} (g x)^m \log \left (e+f x^m\right )}{f g m^2}-\frac {b e k n x^{-m} (g x)^m \log \left (-\frac {f x^m}{e}\right ) \log \left (e+f x^m\right )}{f g m^2}+\frac {2 b k n (g x)^m}{g m^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 16
Rule 19
Rule 20
Rule 32
Rule 45
Rule 272
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 {k (g x)^m \left (a+b \log \left (c x^n\right )\right )}{g m}+\frac {e k x^{-m} (g x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{f 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 {k (g x)^m}{g m x}+\frac {e k x^{-1-m} (g x)^m \log \left (e+f x^m\right )}{f g m}+\frac {(g x)^m \log \left (d \left (e+f x^m\right )^k\right )}{g m x}\right ) \, dx\\ &=-\frac {k (g x)^m \left (a+b \log \left (c x^n\right )\right )}{g m}+\frac {e k x^{-m} (g x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{f 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 \frac {(g x)^m \log \left (d \left (e+f x^m\right )^k\right )}{x} \, dx}{g m}+\frac {(b k n) \int \frac {(g x)^m}{x} \, dx}{g m}-\frac {(b e k n) \int x^{-1-m} (g x)^m \log \left (e+f x^m\right ) \, dx}{f g m}\\ &=-\frac {k (g x)^m \left (a+b \log \left (c x^n\right )\right )}{g m}+\frac {e k x^{-m} (g x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{f 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 {(b k n) \int (g x)^{-1+m} \, dx}{m}-\frac {\left (b e k n x^{-m} (g x)^m\right ) \int \frac {\log \left (e+f x^m\right )}{x} \, dx}{f g m}\\ &=\frac {b k n (g x)^m}{g m^2}-\frac {k (g x)^m \left (a+b \log \left (c x^n\right )\right )}{g m}+\frac {e k x^{-m} (g x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{f 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 e k n x^{-m} (g x)^m\right ) \text {Subst}\left (\int \frac {\log (e+f x)}{x} \, dx,x,x^m\right )}{f g m^2}\\ &=\frac {b k n (g x)^m}{g m^2}-\frac {k (g x)^m \left (a+b \log \left (c x^n\right )\right )}{g m}-\frac {b e k n x^{-m} (g x)^m \log \left (-\frac {f x^m}{e}\right ) \log \left (e+f x^m\right )}{f g m^2}+\frac {e k x^{-m} (g x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{f 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 e 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 )}{g m^2}+\frac {\left (b f k n x^{-m} (g x)^m\right ) \int \frac {x^{-1+2 m}}{e+f x^m} \, dx}{g m}\\ &=\frac {b k n (g x)^m}{g m^2}-\frac {k (g x)^m \left (a+b \log \left (c x^n\right )\right )}{g m}-\frac {b e k n x^{-m} (g x)^m \log \left (-\frac {f x^m}{e}\right ) \log \left (e+f x^m\right )}{f g m^2}+\frac {e k x^{-m} (g x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{f 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 e k n x^{-m} (g x)^m \text {Li}_2\left (1+\frac {f x^m}{e}\right )}{f g m^2}+\frac {\left (b f k n x^{-m} (g x)^m\right ) \text {Subst}\left (\int \frac {x}{e+f x} \, dx,x,x^m\right )}{g m^2}\\ &=\frac {b k n (g x)^m}{g m^2}-\frac {k (g x)^m \left (a+b \log \left (c x^n\right )\right )}{g m}-\frac {b e k n x^{-m} (g x)^m \log \left (-\frac {f x^m}{e}\right ) \log \left (e+f x^m\right )}{f g m^2}+\frac {e k x^{-m} (g x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{f 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 e k n x^{-m} (g x)^m \text {Li}_2\left (1+\frac {f x^m}{e}\right )}{f g m^2}+\frac {\left (b f k n x^{-m} (g x)^m\right ) \text {Subst}\left (\int \left (\frac {1}{f}-\frac {e}{f (e+f x)}\right ) \, dx,x,x^m\right )}{g m^2}\\ &=\frac {2 b k n (g x)^m}{g m^2}-\frac {k (g x)^m \left (a+b \log \left (c x^n\right )\right )}{g m}-\frac {b e k n x^{-m} (g x)^m \log \left (e+f x^m\right )}{f g m^2}-\frac {b e k n x^{-m} (g x)^m \log \left (-\frac {f x^m}{e}\right ) \log \left (e+f x^m\right )}{f g m^2}+\frac {e k x^{-m} (g x)^m \left (a+b \log \left (c x^n\right )\right ) \log \left (e+f x^m\right )}{f 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 e k n x^{-m} (g x)^m \text {Li}_2\left (1+\frac {f x^m}{e}\right )}{f g m^2}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 268, normalized size = 1.05 \begin {gather*} -\frac {x^{-m} (g x)^m \left (a f k m x^m-2 b f k n x^m+b e k m^2 n \log ^2(x)+b f k m x^m \log \left (c x^n\right )-a e k m \log \left (e-e x^m\right )+b e k n \log \left (e-e x^m\right )-b e k m \log \left (c x^n\right ) \log \left (e-e x^m\right )+b e k n \log \left (-\frac {f x^m}{e}\right ) \log \left (e+f x^m\right )-e k m \log (x) \left (a m-b n+b m \log \left (c x^n\right )-b n \log \left (e-e x^m\right )+b n \log \left (e+f x^m\right )\right )-a f m x^m \log \left (d \left (e+f x^m\right )^k\right )+b f n x^m \log \left (d \left (e+f x^m\right )^k\right )-b f m x^m \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )+b e k n \text {Li}_2\left (1+\frac {f x^m}{e}\right )\right )}{f 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 \[\int \left (g x \right )^{-1+m} \left (a +b \ln \left (c \,x^{n}\right )\right ) \ln \left (d \left (e +f \,x^{m}\right )^{k}\right )\, dx\]
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 = 0.36, size = 202, normalized size = 0.79 \begin {gather*} \frac {b g^{m - 1} k m n e \log \left ({\left (f x^{m} + e\right )} e^{\left (-1\right )}\right ) \log \left (x\right ) + b g^{m - 1} k n {\rm Li}_2\left (-{\left (f x^{m} + e\right )} e^{\left (-1\right )} + 1\right ) e - {\left (b f k m \log \left (c\right ) + a f k m - 2 \, b f k n - {\left (b f m \log \left (c\right ) + a f m - b f n\right )} \log \left (d\right ) + {\left (b f k m n - b f m n \log \left (d\right )\right )} \log \left (x\right )\right )} g^{m - 1} x^{m} + {\left ({\left (b f k m n \log \left (x\right ) + 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 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 )}{f m^{2}} \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 \ln \left (d\,{\left (e+f\,x^m\right )}^k\right )\,{\left (g\,x\right )}^{m-1}\,\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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