Optimal. Leaf size=45 \[ \frac{(f+g x) \log \left (c \left (d+\frac{e}{f+g x}\right )^q\right )}{g}+\frac{e q \log (d (f+g x)+e)}{d g} \]
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Rubi [A] time = 0.0267869, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2483, 2448, 263, 31} \[ \frac{(f+g x) \log \left (c \left (d+\frac{e}{f+g x}\right )^q\right )}{g}+\frac{e q \log (d (f+g x)+e)}{d g} \]
Antiderivative was successfully verified.
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Rule 2483
Rule 2448
Rule 263
Rule 31
Rubi steps
\begin{align*} \int \log \left (c \left (d+\frac{e}{f+g x}\right )^q\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \log \left (c \left (d+\frac{e}{x}\right )^q\right ) \, dx,x,f+g x\right )}{g}\\ &=\frac{(f+g x) \log \left (c \left (d+\frac{e}{f+g x}\right )^q\right )}{g}+\frac{(e q) \operatorname{Subst}\left (\int \frac{1}{\left (d+\frac{e}{x}\right ) x} \, dx,x,f+g x\right )}{g}\\ &=\frac{(f+g x) \log \left (c \left (d+\frac{e}{f+g x}\right )^q\right )}{g}+\frac{(e q) \operatorname{Subst}\left (\int \frac{1}{e+d x} \, dx,x,f+g x\right )}{g}\\ &=\frac{(f+g x) \log \left (c \left (d+\frac{e}{f+g x}\right )^q\right )}{g}+\frac{e q \log (e+d (f+g x))}{d g}\\ \end{align*}
Mathematica [A] time = 0.0468237, size = 56, normalized size = 1.24 \[ \frac{d g x \log \left (c \left (d+\frac{e}{f+g x}\right )^q\right )+q (d f+e) \log (d f+d g x+e)-d f q \log (f+g x)}{d g} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.131, size = 74, normalized size = 1.6 \begin{align*} \ln \left ( c \left ({\frac{dgx+df+e}{gx+f}} \right ) ^{q} \right ) x-{\frac{qf\ln \left ( gx+f \right ) }{g}}+{\frac{q\ln \left ( dgx+df+e \right ) f}{g}}+{\frac{eq\ln \left ( dgx+df+e \right ) }{dg}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02127, size = 88, normalized size = 1.96 \begin{align*} -e g q{\left (\frac{f \log \left (g x + f\right )}{e g^{2}} - \frac{{\left (d f + e\right )} \log \left (d g x + d f + e\right )}{d e g^{2}}\right )} + x \log \left (c{\left (d + \frac{e}{g x + f}\right )}^{q}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.602, size = 163, normalized size = 3.62 \begin{align*} \frac{d g q x \log \left (\frac{d g x + d f + e}{g x + f}\right ) - d f q \log \left (g x + f\right ) + d g x \log \left (c\right ) +{\left (d f + e\right )} q \log \left (d g x + d f + e\right )}{d g} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.37966, size = 109, normalized size = 2.42 \begin{align*} \begin{cases} x \log{\left (c \left (\frac{e}{f}\right )^{q} \right )} & \text{for}\: d = 0 \wedge g = 0 \\x \log{\left (c \left (d + \frac{e}{f}\right )^{q} \right )} & \text{for}\: g = 0 \\- \frac{f q \log{\left (f + g x \right )}}{g} + q x \log{\left (e \right )} - q x \log{\left (f + g x \right )} + q x + x \log{\left (c \right )} & \text{for}\: d = 0 \\\frac{f q \log{\left (d + \frac{e}{f + g x} \right )}}{g} + q x \log{\left (d + \frac{e}{f + g x} \right )} + x \log{\left (c \right )} + \frac{e q \log{\left (d f + d g x + e \right )}}{d g} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.30012, size = 116, normalized size = 2.58 \begin{align*} \frac{d g q x \log \left (d g x + d f + e\right ) - d g q x \log \left (g x + f\right ) + d f q \log \left (d g x + d f + e\right ) - d f q \log \left (-g x - f\right ) + d g x \log \left (c\right ) + q e \log \left (d g x + d f + e\right )}{d g} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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