Optimal. Leaf size=677 \[ \text{result too large to display} \]
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Rubi [A] time = 0.981997, antiderivative size = 677, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {2454, 2401, 2389, 2300, 2181, 2390, 2310} \[ \frac{5 d^2 2^{-p} e^{-\frac{2 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )\right )}{b}\right )}{c^2 e^6}+\frac{3^{-p-1} e^{-\frac{3 a}{b}} \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{3 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )\right )}{b}\right )}{c^3 e^6}-\frac{5 d^3 2^{p+2} 3^{-p-1} e^{-\frac{3 a}{2 b}} \left (d+e \sqrt{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{3 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )\right )}{2 b}\right )}{e^6 \left (c \left (d+e \sqrt{x}\right )^2\right )^{3/2}}+\frac{5 d^4 e^{-\frac{a}{b}} \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )}{b}\right )}{c e^6}-\frac{d^5 2^{p+1} e^{-\frac{a}{2 b}} \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )}{2 b}\right )}{e^6 \sqrt{c \left (d+e \sqrt{x}\right )^2}}-\frac{d 2^{p+1} 5^{-p} e^{-\frac{5 a}{2 b}} \left (d+e \sqrt{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )}{b}\right )^{-p} \text{Gamma}\left (p+1,-\frac{5 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )\right )}{2 b}\right )}{e^6 \left (c \left (d+e \sqrt{x}\right )^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2454
Rule 2401
Rule 2389
Rule 2300
Rule 2181
Rule 2390
Rule 2310
Rubi steps
\begin{align*} \int x^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )\right )^p \, dx &=2 \operatorname{Subst}\left (\int x^5 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-\frac{d^5 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^5}+\frac{5 d^4 (d+e x) \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^5}-\frac{10 d^3 (d+e x)^2 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^5}+\frac{10 d^2 (d+e x)^3 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^5}-\frac{5 d (d+e x)^4 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^5}+\frac{(d+e x)^5 \left (a+b \log \left (c (d+e x)^2\right )\right )^p}{e^5}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{2 \operatorname{Subst}\left (\int (d+e x)^5 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\sqrt{x}\right )}{e^5}-\frac{(10 d) \operatorname{Subst}\left (\int (d+e x)^4 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\sqrt{x}\right )}{e^5}+\frac{\left (20 d^2\right ) \operatorname{Subst}\left (\int (d+e x)^3 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\sqrt{x}\right )}{e^5}-\frac{\left (20 d^3\right ) \operatorname{Subst}\left (\int (d+e x)^2 \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\sqrt{x}\right )}{e^5}+\frac{\left (10 d^4\right ) \operatorname{Subst}\left (\int (d+e x) \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\sqrt{x}\right )}{e^5}-\frac{\left (2 d^5\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c (d+e x)^2\right )\right )^p \, dx,x,\sqrt{x}\right )}{e^5}\\ &=\frac{2 \operatorname{Subst}\left (\int x^5 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+e \sqrt{x}\right )}{e^6}-\frac{(10 d) \operatorname{Subst}\left (\int x^4 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+e \sqrt{x}\right )}{e^6}+\frac{\left (20 d^2\right ) \operatorname{Subst}\left (\int x^3 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+e \sqrt{x}\right )}{e^6}-\frac{\left (20 d^3\right ) \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+e \sqrt{x}\right )}{e^6}+\frac{\left (10 d^4\right ) \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+e \sqrt{x}\right )}{e^6}-\frac{\left (2 d^5\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^2\right )\right )^p \, dx,x,d+e \sqrt{x}\right )}{e^6}\\ &=\frac{\operatorname{Subst}\left (\int e^{3 x} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt{x}\right )^2\right )\right )}{c^3 e^6}+\frac{\left (10 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 )^2\right )\right )}{c^2 e^6}+\frac{\left (5 d^4\right ) \operatorname{Subst}\left (\int e^x (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt{x}\right )^2\right )\right )}{c e^6}-\frac{\left (5 d \left (d+e \sqrt{x}\right )^5\right ) \operatorname{Subst}\left (\int e^{5 x/2} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt{x}\right )^2\right )\right )}{e^6 \left (c \left (d+e \sqrt{x}\right )^2\right )^{5/2}}-\frac{\left (10 d^3 \left (d+e \sqrt{x}\right )^3\right ) \operatorname{Subst}\left (\int e^{3 x/2} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt{x}\right )^2\right )\right )}{e^6 \left (c \left (d+e \sqrt{x}\right )^2\right )^{3/2}}-\frac{\left (d^5 \left (d+e \sqrt{x}\right )\right ) \operatorname{Subst}\left (\int e^{x/2} (a+b x)^p \, dx,x,\log \left (c \left (d+e \sqrt{x}\right )^2\right )\right )}{e^6 \sqrt{c \left (d+e \sqrt{x}\right )^2}}\\ &=\frac{3^{-1-p} e^{-\frac{3 a}{b}} \Gamma \left (1+p,-\frac{3 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )}{b}\right )^{-p}}{c^3 e^6}-\frac{2^{1+p} 5^{-p} d e^{-\frac{5 a}{2 b}} \left (d+e \sqrt{x}\right )^5 \Gamma \left (1+p,-\frac{5 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )}{b}\right )^{-p}}{e^6 \left (c \left (d+e \sqrt{x}\right )^2\right )^{5/2}}+\frac{5\ 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 )^2\right )\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )}{b}\right )^{-p}}{c^2 e^6}-\frac{5\ 2^{2+p} 3^{-1-p} d^3 e^{-\frac{3 a}{2 b}} \left (d+e \sqrt{x}\right )^3 \Gamma \left (1+p,-\frac{3 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )}{b}\right )^{-p}}{e^6 \left (c \left (d+e \sqrt{x}\right )^2\right )^{3/2}}+\frac{5 d^4 e^{-\frac{a}{b}} \Gamma \left (1+p,-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )}{b}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )}{b}\right )^{-p}}{c e^6}-\frac{2^{1+p} d^5 e^{-\frac{a}{2 b}} \left (d+e \sqrt{x}\right ) \Gamma \left (1+p,-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )}{2 b}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )\right )^p \left (-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )}{b}\right )^{-p}}{e^6 \sqrt{c \left (d+e \sqrt{x}\right )^2}}\\ \end{align*}
Mathematica [F] time = 0.326332, size = 0, normalized size = 0. \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^2\right )\right )^p \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.088, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( a+b\ln \left ( c \left ( d+e\sqrt{x} \right ) ^{2} \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 )}^{2} c\right ) + a\right )}^{p} x^{2}\,{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^{2} x + 2 \, c d e \sqrt{x} + c d^{2}\right ) + a\right )}^{p} x^{2}, 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 )}^{2} c\right ) + a\right )}^{p} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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