Optimal. Leaf size=131 \[ \frac{(e+f x) e^{-\frac{a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{1}{p q}} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \left (-\frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )^{-n} \text{Gamma}\left (n+1,-\frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )}{f} \]
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Rubi [A] time = 0.151447, 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} \[ \frac{(e+f x) e^{-\frac{a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{1}{p q}} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \left (-\frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )^{-n} \text{Gamma}\left (n+1,-\frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )}{f} \]
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)^p\right )^q\right )\right )^n \, dx &=\operatorname{Subst}\left (\int \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^n \, 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 \left (a+b \log \left (c d^q x^{p q}\right )\right )^n \, 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 )\\ &=\operatorname{Subst}\left (\frac{\left ((e+f x) \left (c d^q (e+f x)^{p q}\right )^{-\frac{1}{p q}}\right ) \operatorname{Subst}\left (\int e^{\frac{x}{p q}} (a+b x)^n \, dx,x,\log \left (c d^q (e+f x)^{p q}\right )\right )}{f p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{e^{-\frac{a}{b p q}} (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{1}{p q}} \Gamma \left (1+n,-\frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \left (-\frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )^{-n}}{f}\\ \end{align*}
Mathematica [A] time = 0.121365, size = 131, normalized size = 1. \[ \frac{(e+f x) e^{-\frac{a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{1}{p q}} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \left (-\frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )^{-n} \text{Gamma}\left (n+1,-\frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )}{f} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.278, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{p} \right ) ^{q} \right ) \right ) ^{n}\, 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 = 2.02874, size = 192, normalized size = 1.47 \begin{align*} \frac{e^{\left (-\frac{b n p q \log \left (-\frac{1}{b p q}\right ) + b q \log \left (d\right ) + b \log \left (c\right ) + a}{b p q}\right )} \Gamma \left (n + 1, -\frac{b p q \log \left (f x + e\right ) + b q \log \left (d\right ) + b \log \left (c\right ) + a}{b p q}\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 )^{p}\right )^{q} \right )}\right )^{n}\, 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 )}^{p} d\right )^{q} c\right ) + a\right )}^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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