Optimal. Leaf size=69 \[ \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d f m \left (d+e x^m\right )}-\frac {b n x^{-m} (f x)^m \log \left (d+e x^m\right )}{d e f m^2} \]
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Rubi [A] time = 0.11, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2335, 268, 260} \[ \frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d f m \left (d+e x^m\right )}-\frac {b n x^{-m} (f x)^m \log \left (d+e x^m\right )}{d e f m^2} \]
Antiderivative was successfully verified.
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Rule 260
Rule 268
Rule 2335
Rubi steps
\begin {align*} \int \frac {(f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^m\right )^2} \, dx &=\frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d f m \left (d+e x^m\right )}-\frac {(b n) \int \frac {(f x)^{-1+m}}{d+e x^m} \, dx}{d m}\\ &=\frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d f m \left (d+e x^m\right )}-\frac {\left (b n x^{-m} (f x)^m\right ) \int \frac {x^{-1+m}}{d+e x^m} \, dx}{d f m}\\ &=\frac {(f x)^m \left (a+b \log \left (c x^n\right )\right )}{d f m \left (d+e x^m\right )}-\frac {b n x^{-m} (f x)^m \log \left (d+e x^m\right )}{d e f m^2}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 89, normalized size = 1.29 \[ -\frac {x^{-m} (f x)^m \left (a d m+b d m \log \left (c x^n\right )+b e n x^m \log \left (d+e x^m\right )-b m n \log (x) \left (d+e x^m\right )+b d n \log \left (d+e x^m\right )\right )}{d e f m^2 \left (d+e x^m\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 89, normalized size = 1.29 \[ \frac {b e f^{m - 1} m n x^{m} \log \relax (x) - {\left (b d m \log \relax (c) + a d m\right )} f^{m - 1} - {\left (b e f^{m - 1} n x^{m} + b d f^{m - 1} n\right )} \log \left (e x^{m} + d\right )}{d e^{2} m^{2} x^{m} + d^{2} e m^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.45, size = 206, normalized size = 2.99 \[ \frac {b f^{m} m n x x^{m} e \log \relax (x)}{d f m^{2} x x^{m} e^{2} + d^{2} f m^{2} x e} - \frac {b f^{m} n x x^{m} e \log \left (x^{m} e + d\right )}{d f m^{2} x x^{m} e^{2} + d^{2} f m^{2} x e} - \frac {b d f^{m} n x \log \left (x^{m} e + d\right )}{d f m^{2} x x^{m} e^{2} + d^{2} f m^{2} x e} - \frac {b d f^{m} m x \log \relax (c)}{d f m^{2} x x^{m} e^{2} + d^{2} f m^{2} x e} - \frac {a d f^{m} m x}{d f m^{2} x x^{m} e^{2} + d^{2} f m^{2} x e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.27, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right ) \left (f x \right )^{m -1}}{\left (e \,x^{m}+d \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.72, size = 97, normalized size = 1.41 \[ b f^{m} n {\left (\frac {\log \relax (x)}{d e f m} - \frac {\log \left (e x^{m} + d\right )}{d e f m^{2}}\right )} - \frac {b f^{m} \log \left (c x^{n}\right )}{e^{2} f m x^{m} + d e f m} - \frac {a f^{m}}{e^{2} f m x^{m} + d e f m} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (f\,x\right )}^{m-1}\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (d+e\,x^m\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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