Optimal. Leaf size=50 \[ a x+\frac{b (f+g x) \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )}{g}+\frac{b e p \log (d (f+g x)+e)}{d g} \]
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Rubi [A] time = 0.0376744, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2483, 2448, 263, 31} \[ a x+\frac{b (f+g x) \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )}{g}+\frac{b e p \log (d (f+g x)+e)}{d g} \]
Antiderivative was successfully verified.
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Rule 2483
Rule 2448
Rule 263
Rule 31
Rubi steps
\begin{align*} \int \left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right ) \, dx &=a x+b \int \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right ) \, dx\\ &=a x+\frac{b \operatorname{Subst}\left (\int \log \left (c \left (d+\frac{e}{x}\right )^p\right ) \, dx,x,f+g x\right )}{g}\\ &=a x+\frac{b (f+g x) \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )}{g}+\frac{(b e p) \operatorname{Subst}\left (\int \frac{1}{\left (d+\frac{e}{x}\right ) x} \, dx,x,f+g x\right )}{g}\\ &=a x+\frac{b (f+g x) \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )}{g}+\frac{(b e p) \operatorname{Subst}\left (\int \frac{1}{e+d x} \, dx,x,f+g x\right )}{g}\\ &=a x+\frac{b (f+g x) \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )}{g}+\frac{b e p \log (e+d (f+g x))}{d g}\\ \end{align*}
Mathematica [A] time = 0.0408769, size = 70, normalized size = 1.4 \[ a x+b x \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )-b e g p \left (\frac{f \log (f+g x)}{e g^2}-\frac{(d f+e) \log (d f+d g x+e)}{d e g^2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.106, size = 81, normalized size = 1.6 \begin{align*} ax+b\ln \left ( c \left ({\frac{dgx+df+e}{gx+f}} \right ) ^{p} \right ) x-{\frac{bpf\ln \left ( gx+f \right ) }{g}}+{\frac{bp\ln \left ( dgx+df+e \right ) f}{g}}+{\frac{ebp\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 = 1.02494, size = 95, normalized size = 1.9 \begin{align*} -b e g p{\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 )} + b x \log \left (c{\left (d + \frac{e}{g x + f}\right )}^{p}\right ) + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.94278, size = 190, normalized size = 3.8 \begin{align*} \frac{b d g p x \log \left (\frac{d g x + d f + e}{g x + f}\right ) - b d f p \log \left (g x + f\right ) + b d g x \log \left (c\right ) + a d g x +{\left (b d f + b e\right )} p \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 = 2.29712, size = 114, normalized size = 2.28 \begin{align*} a x + b \left (\begin{cases} x \log{\left (c \left (\frac{e}{f}\right )^{p} \right )} & \text{for}\: d = 0 \wedge g = 0 \\x \log{\left (c \left (d + \frac{e}{f}\right )^{p} \right )} & \text{for}\: g = 0 \\- \frac{f p \log{\left (f + g x \right )}}{g} + p x \log{\left (e \right )} - p x \log{\left (f + g x \right )} + p x + x \log{\left (c \right )} & \text{for}\: d = 0 \\\frac{f p \log{\left (d + \frac{e}{f + g x} \right )}}{g} + p x \log{\left (d + \frac{e}{f + g x} \right )} + x \log{\left (c \right )} + \frac{e p \log{\left (d f + d g x + e \right )}}{d g} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31974, size = 123, normalized size = 2.46 \begin{align*} a x + \frac{{\left (d g p x \log \left (d g x + d f + e\right ) - d g p x \log \left (g x + f\right ) + d f p \log \left (d g x + d f + e\right ) - d f p \log \left (-g x - f\right ) + d g x \log \left (c\right ) + p e \log \left (d g x + d f + e\right )\right )} b}{d g} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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