Optimal. Leaf size=38 \[ \frac{(f x)^m \left (a+b \log \left (c x^n\right )\right )}{f m}-\frac{b n (f x)^m}{f m^2} \]
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Rubi [A] time = 0.0173125, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {2304} \[ \frac{(f x)^m \left (a+b \log \left (c x^n\right )\right )}{f m}-\frac{b n (f x)^m}{f m^2} \]
Antiderivative was successfully verified.
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Rule 2304
Rubi steps
\begin{align*} \int (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right ) \, dx &=-\frac{b n (f x)^m}{f m^2}+\frac{(f x)^m \left (a+b \log \left (c x^n\right )\right )}{f m}\\ \end{align*}
Mathematica [A] time = 0.0094612, size = 29, normalized size = 0.76 \[ \frac{(f x)^m \left (a m+b m \log \left (c x^n\right )-b n\right )}{f m^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.102, size = 281, normalized size = 7.4 \begin{align*}{\frac{bx\ln \left ({x}^{n} \right ) }{m}{{\rm e}^{{\frac{ \left ( -1+m \right ) \left ( -i\pi \, \left ({\it csgn} \left ( ifx \right ) \right ) ^{3}+i\pi \, \left ({\it csgn} \left ( ifx \right ) \right ) ^{2}{\it csgn} \left ( if \right ) +i\pi \, \left ({\it csgn} \left ( ifx \right ) \right ) ^{2}{\it csgn} \left ( ix \right ) -i\pi \,{\it csgn} \left ( ifx \right ){\it csgn} \left ( if \right ){\it csgn} \left ( ix \right ) +2\,\ln \left ( f \right ) +2\,\ln \left ( x \right ) \right ) }{2}}}}}+{\frac{ \left ( i\pi \,b{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}m-i\pi \,b{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) m-i\pi \,b \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}m+i\pi \,b \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) m+2\,b\ln \left ( c \right ) m+2\,am-2\,bn \right ) x}{2\,{m}^{2}}{{\rm e}^{{\frac{ \left ( -1+m \right ) \left ( -i\pi \, \left ({\it csgn} \left ( ifx \right ) \right ) ^{3}+i\pi \, \left ({\it csgn} \left ( ifx \right ) \right ) ^{2}{\it csgn} \left ( if \right ) +i\pi \, \left ({\it csgn} \left ( ifx \right ) \right ) ^{2}{\it csgn} \left ( ix \right ) -i\pi \,{\it csgn} \left ( ifx \right ){\it csgn} \left ( if \right ){\it csgn} \left ( ix \right ) +2\,\ln \left ( f \right ) +2\,\ln \left ( x \right ) \right ) }{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.27176, size = 120, normalized size = 3.16 \begin{align*} \frac{{\left (b m n x \log \left (x\right ) + b m x \log \left (c\right ) +{\left (a m - b n\right )} x\right )} e^{\left ({\left (m - 1\right )} \log \left (f\right ) +{\left (m - 1\right )} \log \left (x\right )\right )}}{m^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.3117, size = 86, normalized size = 2.26 \begin{align*} \frac{b f^{m} n x^{m} \log \left (x\right )}{f m} + \frac{b f^{m} x^{m} \log \left (c\right )}{f m} + \frac{a f^{m} x^{m}}{f m} - \frac{b f^{m} n x^{m}}{f m^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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