Optimal. Leaf size=56 \[ -\frac{1}{3} \left (-x^2+2 x+8\right )^{3/2}-\frac{1}{2} (1-x) \sqrt{-x^2+2 x+8}-\frac{9}{2} \sin ^{-1}\left (\frac{1-x}{3}\right ) \]
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Rubi [A] time = 0.0137811, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {640, 612, 619, 216} \[ -\frac{1}{3} \left (-x^2+2 x+8\right )^{3/2}-\frac{1}{2} (1-x) \sqrt{-x^2+2 x+8}-\frac{9}{2} \sin ^{-1}\left (\frac{1-x}{3}\right ) \]
Antiderivative was successfully verified.
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Rule 640
Rule 612
Rule 619
Rule 216
Rubi steps
\begin{align*} \int x \sqrt{8+2 x-x^2} \, dx &=-\frac{1}{3} \left (8+2 x-x^2\right )^{3/2}+\int \sqrt{8+2 x-x^2} \, dx\\ &=-\frac{1}{2} (1-x) \sqrt{8+2 x-x^2}-\frac{1}{3} \left (8+2 x-x^2\right )^{3/2}+\frac{9}{2} \int \frac{1}{\sqrt{8+2 x-x^2}} \, dx\\ &=-\frac{1}{2} (1-x) \sqrt{8+2 x-x^2}-\frac{1}{3} \left (8+2 x-x^2\right )^{3/2}-\frac{3}{4} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{36}}} \, dx,x,2-2 x\right )\\ &=-\frac{1}{2} (1-x) \sqrt{8+2 x-x^2}-\frac{1}{3} \left (8+2 x-x^2\right )^{3/2}-\frac{9}{2} \sin ^{-1}\left (\frac{1-x}{3}\right )\\ \end{align*}
Mathematica [A] time = 0.0196784, size = 42, normalized size = 0.75 \[ \frac{1}{6} \left (\sqrt{-x^2+2 x+8} \left (2 x^2-x-19\right )-27 \sin ^{-1}\left (\frac{1}{3}-\frac{x}{3}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 43, normalized size = 0.8 \begin{align*} -{\frac{1}{3} \left ( -{x}^{2}+2\,x+8 \right ) ^{{\frac{3}{2}}}}-{\frac{2-2\,x}{4}\sqrt{-{x}^{2}+2\,x+8}}+{\frac{9}{2}\arcsin \left ( -{\frac{1}{3}}+{\frac{x}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.47808, size = 70, normalized size = 1.25 \begin{align*} -\frac{1}{3} \,{\left (-x^{2} + 2 \, x + 8\right )}^{\frac{3}{2}} + \frac{1}{2} \, \sqrt{-x^{2} + 2 \, x + 8} x - \frac{1}{2} \, \sqrt{-x^{2} + 2 \, x + 8} - \frac{9}{2} \, \arcsin \left (-\frac{1}{3} \, x + \frac{1}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.03887, size = 138, normalized size = 2.46 \begin{align*} \frac{1}{6} \,{\left (2 \, x^{2} - x - 19\right )} \sqrt{-x^{2} + 2 \, x + 8} - \frac{9}{2} \, \arctan \left (\frac{\sqrt{-x^{2} + 2 \, x + 8}{\left (x - 1\right )}}{x^{2} - 2 \, x - 8}\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 x \sqrt{- \left (x - 4\right ) \left (x + 2\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09454, size = 43, normalized size = 0.77 \begin{align*} \frac{1}{6} \,{\left ({\left (2 \, x - 1\right )} x - 19\right )} \sqrt{-x^{2} + 2 \, x + 8} + \frac{9}{2} \, \arcsin \left (\frac{1}{3} \, x - \frac{1}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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