Optimal. Leaf size=360 \[ \frac{3 d^2 2^{-p} e^{-\frac{2 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt{x}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )\right )\right )}{b}\right )}{c^2 e^4}+\frac{2^{-2 p-1} e^{-\frac{4 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt{x}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{4 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )\right )\right )}{b}\right )}{c^4 e^4}-\frac{2 d 3^{-p} e^{-\frac{3 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt{x}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{3 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )\right )\right )}{b}\right )}{c^3 e^4}-\frac{2 d^3 e^{-\frac{a}{b}} \left (a+b \log \left (c \left (d+e \sqrt{x}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )\right )}{b}\right )}{c e^4} \]
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Rubi [A] time = 0.543432, antiderivative size = 360, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {2454, 2401, 2389, 2299, 2181, 2390, 2309} \[ \frac{3 d^2 2^{-p} e^{-\frac{2 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt{x}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )\right )\right )}{b}\right )}{c^2 e^4}+\frac{2^{-2 p-1} e^{-\frac{4 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt{x}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{4 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )\right )\right )}{b}\right )}{c^4 e^4}-\frac{2 d 3^{-p} e^{-\frac{3 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt{x}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{3 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )\right )\right )}{b}\right )}{c^3 e^4}-\frac{2 d^3 e^{-\frac{a}{b}} \left (a+b \log \left (c \left (d+e \sqrt{x}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )\right )}{b}\right )}{c e^4} \]
Antiderivative was successfully verified.
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Rule 2454
Rule 2401
Rule 2389
Rule 2299
Rule 2181
Rule 2390
Rule 2309
Rubi steps
\begin{align*} \int x \left (a+b \log \left (c \left (d+e \sqrt{x}\right )\right )\right )^p \, dx &=2 \operatorname{Subst}\left (\int x^3 (a+b \log (c (d+e x)))^p \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-\frac{d^3 (a+b \log (c (d+e x)))^p}{e^3}+\frac{3 d^2 (d+e x) (a+b \log (c (d+e x)))^p}{e^3}-\frac{3 d (d+e x)^2 (a+b \log (c (d+e x)))^p}{e^3}+\frac{(d+e x)^3 (a+b \log (c (d+e x)))^p}{e^3}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{2 \operatorname{Subst}\left (\int (d+e x)^3 (a+b \log (c (d+e x)))^p \, dx,x,\sqrt{x}\right )}{e^3}-\frac{(6 d) \operatorname{Subst}\left (\int (d+e x)^2 (a+b \log (c (d+e x)))^p \, dx,x,\sqrt{x}\right )}{e^3}+\frac{\left (6 d^2\right ) \operatorname{Subst}\left (\int (d+e x) (a+b \log (c (d+e x)))^p \, dx,x,\sqrt{x}\right )}{e^3}-\frac{\left (2 d^3\right ) \operatorname{Subst}\left (\int (a+b \log (c (d+e x)))^p \, dx,x,\sqrt{x}\right )}{e^3}\\ &=\frac{2 \operatorname{Subst}\left (\int x^3 (a+b \log (c x))^p \, dx,x,d+e \sqrt{x}\right )}{e^4}-\frac{(6 d) \operatorname{Subst}\left (\int x^2 (a+b \log (c x))^p \, dx,x,d+e \sqrt{x}\right )}{e^4}+\frac{\left (6 d^2\right ) \operatorname{Subst}\left (\int x (a+b \log (c x))^p \, dx,x,d+e \sqrt{x}\right )}{e^4}-\frac{\left (2 d^3\right ) \operatorname{Subst}\left (\int (a+b \log (c x))^p \, dx,x,d+e \sqrt{x}\right )}{e^4}\\ &=\frac{2 \operatorname{Subst}\left (\int e^{4 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt{x}\right )\right )\right )}{c^4 e^4}-\frac{(6 d) \operatorname{Subst}\left (\int e^{3 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt{x}\right )\right )\right )}{c^3 e^4}+\frac{\left (6 d^2\right ) \operatorname{Subst}\left (\int e^{2 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt{x}\right )\right )\right )}{c^2 e^4}-\frac{\left (2 d^3\right ) \operatorname{Subst}\left (\int e^x (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt{x}\right )\right )\right )}{c e^4}\\ &=\frac{2^{-1-2 p} e^{-\frac{4 a}{b}} \Gamma \left (1+p,-\frac{4 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )\right )}{b}\right )^{-p}}{c^4 e^4}-\frac{2\ 3^{-p} d e^{-\frac{3 a}{b}} \Gamma \left (1+p,-\frac{3 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )\right )}{b}\right )^{-p}}{c^3 e^4}+\frac{3\ 2^{-p} d^2 e^{-\frac{2 a}{b}} \Gamma \left (1+p,-\frac{2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )\right )}{b}\right )^{-p}}{c^2 e^4}-\frac{2 d^3 e^{-\frac{a}{b}} \Gamma \left (1+p,-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )\right )}{b}\right )^{-p}}{c e^4}\\ \end{align*}
Mathematica [A] time = 0.421726, size = 229, normalized size = 0.64 \[ \frac{2^{-2 p-1} 3^{-p} e^{-\frac{4 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt{x}\right )\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )\right )}{b}\right )^{-p} \left (3^p \text{Gamma}\left (p+1,-\frac{4 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )\right )\right )}{b}\right )-c d 2^{p+1} e^{a/b} \left (2^{p+1} \text{Gamma}\left (p+1,-\frac{3 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )\right )\right )}{b}\right )+c d 3^p e^{a/b} \left (c d 2^{p+1} e^{a/b} \text{Gamma}\left (p+1,-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )\right )}{b}\right )-3 \text{Gamma}\left (p+1,-\frac{2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )\right )\right )}{b}\right )\right )\right )\right )}{c^4 e^4} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.087, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b\ln \left ( c \left ( d+e\sqrt{x} \right ) \right ) \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left ({\left (e \sqrt{x} + d\right )} c\right ) + a\right )}^{p} x\,{d x} \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 (c e \sqrt{x} + c d\right ) + a\right )}^{p} x, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 \sqrt{x} + d\right )} c\right ) + a\right )}^{p} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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