Optimal. Leaf size=34 \[ a x+\frac{b (e+f x) \log \left (c \left (d (e+f x)^m\right )^n\right )}{f}-b m n x \]
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Rubi [A] time = 0.0315224, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2389, 2295, 2445} \[ a x+\frac{b (e+f x) \log \left (c \left (d (e+f x)^m\right )^n\right )}{f}-b m n x \]
Antiderivative was successfully verified.
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Rule 2389
Rule 2295
Rule 2445
Rubi steps
\begin{align*} \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right ) \, dx &=a x+b \int \log \left (c \left (d (e+f x)^m\right )^n\right ) \, dx\\ &=a x+b \operatorname{Subst}\left (\int \log \left (c d^n (e+f x)^{m n}\right ) \, dx,c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right )\\ &=a x+b \operatorname{Subst}\left (\frac{\operatorname{Subst}\left (\int \log \left (c d^n x^{m n}\right ) \, dx,x,e+f x\right )}{f},c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right )\\ &=a x-b m n x+\frac{b (e+f x) \log \left (c \left (d (e+f x)^m\right )^n\right )}{f}\\ \end{align*}
Mathematica [A] time = 0.006675, size = 34, normalized size = 1. \[ a x+\frac{b (e+f x) \log \left (c \left (d (e+f x)^m\right )^n\right )}{f}-b m n x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.066, size = 42, normalized size = 1.2 \begin{align*} ax+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{m} \right ) ^{n} \right ) x-bmnx+{\frac{bemn\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 = 1.12163, size = 61, normalized size = 1.79 \begin{align*} -b f m n{\left (\frac{x}{f} - \frac{e \log \left (f x + e\right )}{f^{2}}\right )} + b x \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right ) + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.27636, size = 124, normalized size = 3.65 \begin{align*} \frac{b f n x \log \left (d\right ) + b f x \log \left (c\right ) -{\left (b f m n - a f\right )} x +{\left (b f m n x + b e m n\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 = 0.920717, size = 58, normalized size = 1.71 \begin{align*} a x + b \left (\begin{cases} \frac{e m n \log{\left (e + f x \right )}}{f} + m n x \log{\left (e + f x \right )} - m n x + n x \log{\left (d \right )} + x \log{\left (c \right )} & \text{for}\: f \neq 0 \\x \log{\left (c \left (d e^{m}\right )^{n} \right )} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23734, size = 86, normalized size = 2.53 \begin{align*}{\left (\frac{{\left (f x + e\right )} m n \log \left (f x + e\right )}{f} - \frac{{\left (f x + e\right )} m n}{f} + \frac{{\left (f x + e\right )} n \log \left (d\right )}{f} + \frac{{\left (f x + e\right )} \log \left (c\right )}{f}\right )} b + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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