Optimal. Leaf size=29 \[ \frac{(e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f}-p q x \]
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Rubi [A] time = 0.0230847, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {2389, 2295, 2445} \[ \frac{(e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f}-p q x \]
Antiderivative was successfully verified.
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Rule 2389
Rule 2295
Rule 2445
Rubi steps
\begin{align*} \int \log \left (c \left (d (e+f x)^p\right )^q\right ) \, dx &=\operatorname{Subst}\left (\int \log \left (c d^q (e+f x)^{p q}\right ) \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\operatorname{Subst}\left (\frac{\operatorname{Subst}\left (\int \log \left (c d^q x^{p q}\right ) \, dx,x,e+f x\right )}{f},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-p q x+\frac{(e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f}\\ \end{align*}
Mathematica [A] time = 0.0097612, size = 29, normalized size = 1. \[ \frac{(e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f}-p q x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.066, size = 36, normalized size = 1.2 \begin{align*} \ln \left ( c \left ( d \left ( fx+e \right ) ^{p} \right ) ^{q} \right ) x-pqx+{\frac{pqe\ln \left ( fx+e \right ) }{f}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12423, size = 54, normalized size = 1.86 \begin{align*} -f p q{\left (\frac{x}{f} - \frac{e \log \left (f x + e\right )}{f^{2}}\right )} + x \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.95057, size = 101, normalized size = 3.48 \begin{align*} -\frac{f p q x - f q x \log \left (d\right ) - f x \log \left (c\right ) -{\left (f p q x + e p q\right )} \log \left (f x + e\right )}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.917149, size = 53, normalized size = 1.83 \begin{align*} \begin{cases} \frac{e p q \log{\left (e + f x \right )}}{f} + p q x \log{\left (e + f x \right )} - p q x + q x \log{\left (d \right )} + x \log{\left (c \right )} & \text{for}\: f \neq 0 \\x \log{\left (c \left (d e^{p}\right )^{q} \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26604, size = 78, normalized size = 2.69 \begin{align*} \frac{{\left (f x + e\right )} p q \log \left (f x + e\right )}{f} - \frac{{\left (f x + e\right )} p q}{f} + \frac{{\left (f x + e\right )} q \log \left (d\right )}{f} + \frac{{\left (f x + e\right )} \log \left (c\right )}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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