Optimal. Leaf size=109 \[ \frac{4^{-p} e^{-\frac{4 a}{b}} \left (a+b \log \left (c \sqrt{d \sqrt{e+f x}}\right )\right )^p \left (-\frac{a+b \log \left (c \sqrt{d \sqrt{e+f x}}\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{4 \left (a+b \log \left (c \sqrt{d \sqrt{e+f x}}\right )\right )}{b}\right )}{c^4 d^2 f} \]
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Rubi [A] time = 0.138649, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2389, 2299, 2181, 2445} \[ \frac{4^{-p} e^{-\frac{4 a}{b}} \left (a+b \log \left (c \sqrt{d \sqrt{e+f x}}\right )\right )^p \left (-\frac{a+b \log \left (c \sqrt{d \sqrt{e+f x}}\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{4 \left (a+b \log \left (c \sqrt{d \sqrt{e+f x}}\right )\right )}{b}\right )}{c^4 d^2 f} \]
Antiderivative was successfully verified.
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Rule 2389
Rule 2299
Rule 2181
Rule 2445
Rubi steps
\begin{align*} \int \left (a+b \log \left (c \sqrt{d \sqrt{e+f x}}\right )\right )^p \, dx &=\operatorname{Subst}\left (\int \left (a+b \log \left (c \sqrt{d} \sqrt [4]{e+f x}\right )\right )^p \, dx,c \sqrt{d} \sqrt [4]{e+f x},c \sqrt{d \sqrt{e+f x}}\right )\\ &=\operatorname{Subst}\left (\frac{\operatorname{Subst}\left (\int \left (a+b \log \left (c \sqrt{d} \sqrt [4]{x}\right )\right )^p \, dx,x,e+f x\right )}{f},c \sqrt{d} \sqrt [4]{e+f x},c \sqrt{d \sqrt{e+f x}}\right )\\ &=\operatorname{Subst}\left (\frac{4 \operatorname{Subst}\left (\int e^{4 x} (a+b x)^p \, dx,x,\log \left (c \sqrt{d} \sqrt [4]{e+f x}\right )\right )}{c^4 d^2 f},c \sqrt{d} \sqrt [4]{e+f x},c \sqrt{d \sqrt{e+f x}}\right )\\ &=\frac{4^{-p} e^{-\frac{4 a}{b}} \Gamma \left (1+p,-\frac{4 \left (a+b \log \left (c \sqrt{d \sqrt{e+f x}}\right )\right )}{b}\right ) \left (a+b \log \left (c \sqrt{d \sqrt{e+f x}}\right )\right )^p \left (-\frac{a+b \log \left (c \sqrt{d \sqrt{e+f x}}\right )}{b}\right )^{-p}}{c^4 d^2 f}\\ \end{align*}
Mathematica [A] time = 0.10862, size = 109, normalized size = 1. \[ \frac{2^{-2 p} e^{-\frac{4 a}{b}} \left (a+b \log \left (c \sqrt{d \sqrt{e+f x}}\right )\right )^p \left (-\frac{a+b \log \left (c \sqrt{d \sqrt{e+f x}}\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{4 \left (a+b \log \left (c \sqrt{d \sqrt{e+f x}}\right )\right )}{b}\right )}{c^4 d^2 f} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.426, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c\sqrt{d\sqrt{fx+e}} \right ) \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.31926, size = 95, normalized size = 0.87 \begin{align*} -\frac{4 \,{\left (b \log \left (\sqrt{\sqrt{f x + e} d} c\right ) + a\right )}^{p + 1} e^{\left (-\frac{4 \, a}{b}\right )} E_{-p}\left (-\frac{4 \,{\left (b \log \left (\sqrt{\sqrt{f x + e} d} c\right ) + a\right )}}{b}\right )}{b c^{4} d^{2} f} \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{\sqrt{f x + e} 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 \sqrt{e + f 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{\sqrt{f x + e} d} c\right ) + a\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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