Optimal. Leaf size=131 \[ \frac{(e+f x) e^{-\frac{a}{b m n}} \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac{1}{m n}} \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^p \left (-\frac{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}{b m n}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}{b m n}\right )}{f} \]
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Rubi [A] time = 0.14695, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2389, 2300, 2181, 2445} \[ \frac{(e+f x) e^{-\frac{a}{b m n}} \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac{1}{m n}} \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^p \left (-\frac{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}{b m n}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}{b m n}\right )}{f} \]
Antiderivative was successfully verified.
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Rule 2389
Rule 2300
Rule 2181
Rule 2445
Rubi steps
\begin{align*} \int \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^p \, dx &=\operatorname{Subst}\left (\int \left (a+b \log \left (c d^n (e+f x)^{m n}\right )\right )^p \, dx,c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right )\\ &=\operatorname{Subst}\left (\frac{\operatorname{Subst}\left (\int \left (a+b \log \left (c d^n x^{m n}\right )\right )^p \, 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 )\\ &=\operatorname{Subst}\left (\frac{\left ((e+f x) \left (c d^n (e+f x)^{m n}\right )^{-\frac{1}{m n}}\right ) \operatorname{Subst}\left (\int e^{\frac{x}{m n}} (a+b x)^p \, dx,x,\log \left (c d^n (e+f x)^{m n}\right )\right )}{f m n},c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right )\\ &=\frac{e^{-\frac{a}{b m n}} (e+f x) \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac{1}{m n}} \Gamma \left (1+p,-\frac{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}{b m n}\right ) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^p \left (-\frac{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}{b m n}\right )^{-p}}{f}\\ \end{align*}
Mathematica [A] time = 0.163161, size = 131, normalized size = 1. \[ \frac{(e+f x) e^{-\frac{a}{b m n}} \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac{1}{m n}} \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^p \left (-\frac{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}{b m n}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}{b m n}\right )}{f} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.145, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{m} \right ) ^{n} \right ) \right ) ^{p}\, dx \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: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.45504, size = 192, normalized size = 1.47 \begin{align*} \frac{e^{\left (-\frac{b m n p \log \left (-\frac{1}{b m n}\right ) + b n \log \left (d\right ) + b \log \left (c\right ) + a}{b m n}\right )} \Gamma \left (p + 1, -\frac{b m n \log \left (f x + e\right ) + b n \log \left (d\right ) + b \log \left (c\right ) + a}{b m n}\right )}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \log{\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}\right )^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right ) + a\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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