Optimal. Leaf size=103 \[ \frac{e^{-\frac{a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \left (a+b \log \left (c (d+e x)^n\right )\right )^p \left (-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{e} \]
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Rubi [A] time = 0.058868, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {2389, 2300, 2181} \[ \frac{e^{-\frac{a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \left (a+b \log \left (c (d+e x)^n\right )\right )^p \left (-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{e} \]
Antiderivative was successfully verified.
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Rule 2389
Rule 2300
Rule 2181
Rubi steps
\begin{align*} \int \left (a+b \log \left (c (d+e x)^n\right )\right )^p \, dx &=\frac{\operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^p \, dx,x,d+e x\right )}{e}\\ &=\frac{\left ((d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int e^{\frac{x}{n}} (a+b x)^p \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e n}\\ &=\frac{e^{-\frac{a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \Gamma \left (1+p,-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^p \left (-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )^{-p}}{e}\\ \end{align*}
Mathematica [A] time = 0.0937072, size = 103, normalized size = 1. \[ \frac{e^{-\frac{a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \left (a+b \log \left (c (d+e x)^n\right )\right )^p \left (-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{e} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.463, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c \left ( ex+d \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 = 2.23652, size = 143, normalized size = 1.39 \begin{align*} \frac{e^{\left (-\frac{b n p \log \left (-\frac{1}{b n}\right ) + b \log \left (c\right ) + a}{b n}\right )} \Gamma \left (p + 1, -\frac{b n \log \left (e x + d\right ) + b \log \left (c\right ) + a}{b n}\right )}{e} \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 + e x\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 (e x + 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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