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 = 0.107119, antiderivative size = 69, normalized size of antiderivative = 1., 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 2335
Rule 268
Rule 260
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*}
Mathematica [A] time = 0.121435, 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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Maple [F] time = 1.158, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx \right ) ^{-1+m} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }{ \left ( d+e{x}^{m} \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.19838, size = 131, normalized size = 1.9 \begin{align*} b f^{m} n{\left (\frac{\log \left (x\right )}{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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.36141, size = 205, normalized size = 2.97 \begin{align*} \frac{b e f^{m - 1} m n x^{m} \log \left (x\right ) -{\left (b d m \log \left (c\right ) + 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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.36732, size = 278, normalized size = 4.03 \begin{align*} \frac{b f^{m} m n x x^{m} e \log \left (x\right )}{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 \left (c\right )}{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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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