Optimal. Leaf size=88 \[ \frac{2^{-p} e^{-\frac{2 a}{b}} \left (a+b \log \left (c \sqrt{d+e x}\right )\right )^p \left (-\frac{a+b \log \left (c \sqrt{d+e x}\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{2 \left (a+b \log \left (c \sqrt{d+e x}\right )\right )}{b}\right )}{c^2 e} \]
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Rubi [A] time = 0.0675028, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2389, 2299, 2181} \[ \frac{2^{-p} e^{-\frac{2 a}{b}} \left (a+b \log \left (c \sqrt{d+e x}\right )\right )^p \left (-\frac{a+b \log \left (c \sqrt{d+e x}\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{2 \left (a+b \log \left (c \sqrt{d+e x}\right )\right )}{b}\right )}{c^2 e} \]
Antiderivative was successfully verified.
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Rule 2389
Rule 2299
Rule 2181
Rubi steps
\begin{align*} \int \left (a+b \log \left (c \sqrt{d+e x}\right )\right )^p \, dx &=\frac{\operatorname{Subst}\left (\int \left (a+b \log \left (c \sqrt{x}\right )\right )^p \, dx,x,d+e x\right )}{e}\\ &=\frac{2 \operatorname{Subst}\left (\int e^{2 x} (a+b x)^p \, dx,x,\log \left (c \sqrt{d+e x}\right )\right )}{c^2 e}\\ &=\frac{2^{-p} e^{-\frac{2 a}{b}} \Gamma \left (1+p,-\frac{2 \left (a+b \log \left (c \sqrt{d+e x}\right )\right )}{b}\right ) \left (a+b \log \left (c \sqrt{d+e x}\right )\right )^p \left (-\frac{a+b \log \left (c \sqrt{d+e x}\right )}{b}\right )^{-p}}{c^2 e}\\ \end{align*}
Mathematica [A] time = 0.0500074, size = 88, normalized size = 1. \[ \frac{2^{-p} e^{-\frac{2 a}{b}} \left (a+b \log \left (c \sqrt{d+e x}\right )\right )^p \left (-\frac{a+b \log \left (c \sqrt{d+e x}\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{2 \left (a+b \log \left (c \sqrt{d+e x}\right )\right )}{b}\right )}{c^2 e} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.267, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c\sqrt{ex+d} \right ) \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.30587, size = 80, normalized size = 0.91 \begin{align*} -\frac{2 \,{\left (b \log \left (\sqrt{e x + d} c\right ) + a\right )}^{p + 1} e^{\left (-\frac{2 \, a}{b}\right )} E_{-p}\left (-\frac{2 \,{\left (b \log \left (\sqrt{e x + d} c\right ) + a\right )}}{b}\right )}{b c^{2} e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \log \left (\sqrt{e x + d} c\right ) + a\right )}^{p}, x\right ) \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 \sqrt{d + e x} \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 (\sqrt{e x + d} c\right ) + a\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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