Optimal. Leaf size=52 \[ -\frac{1}{3} \left (-x^2-2 x+3\right )^{3/2}-\frac{1}{2} (x+1) \sqrt{-x^2-2 x+3}+2 \sin ^{-1}\left (\frac{1}{2} (-x-1)\right ) \]
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Rubi [A] time = 0.0137335, antiderivative size = 52, 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} \[ -\frac{1}{3} \left (-x^2-2 x+3\right )^{3/2}-\frac{1}{2} (x+1) \sqrt{-x^2-2 x+3}+2 \sin ^{-1}\left (\frac{1}{2} (-x-1)\right ) \]
Antiderivative was successfully verified.
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Rule 640
Rule 612
Rule 619
Rule 216
Rubi steps
\begin{align*} \int x \sqrt{3-2 x-x^2} \, dx &=-\frac{1}{3} \left (3-2 x-x^2\right )^{3/2}-\int \sqrt{3-2 x-x^2} \, dx\\ &=-\frac{1}{2} (1+x) \sqrt{3-2 x-x^2}-\frac{1}{3} \left (3-2 x-x^2\right )^{3/2}-2 \int \frac{1}{\sqrt{3-2 x-x^2}} \, dx\\ &=-\frac{1}{2} (1+x) \sqrt{3-2 x-x^2}-\frac{1}{3} \left (3-2 x-x^2\right )^{3/2}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{16}}} \, dx,x,-2-2 x\right )\\ &=-\frac{1}{2} (1+x) \sqrt{3-2 x-x^2}-\frac{1}{3} \left (3-2 x-x^2\right )^{3/2}+2 \sin ^{-1}\left (\frac{1}{2} (-1-x)\right )\\ \end{align*}
Mathematica [A] time = 0.0174071, size = 37, normalized size = 0.71 \[ \frac{1}{6} \sqrt{-x^2-2 x+3} \left (2 x^2+x-9\right )-2 \sin ^{-1}\left (\frac{x+1}{2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 43, normalized size = 0.8 \begin{align*} -{\frac{1}{3} \left ( -{x}^{2}-2\,x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{-2\,x-2}{4}\sqrt{-{x}^{2}-2\,x+3}}-2\,\arcsin \left ( 1/2+x/2 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49706, size = 70, normalized size = 1.35 \begin{align*} -\frac{1}{3} \,{\left (-x^{2} - 2 \, x + 3\right )}^{\frac{3}{2}} - \frac{1}{2} \, \sqrt{-x^{2} - 2 \, x + 3} x - \frac{1}{2} \, \sqrt{-x^{2} - 2 \, x + 3} + 2 \, \arcsin \left (-\frac{1}{2} \, x - \frac{1}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.08042, size = 134, normalized size = 2.58 \begin{align*} \frac{1}{6} \,{\left (2 \, x^{2} + x - 9\right )} \sqrt{-x^{2} - 2 \, x + 3} + 2 \, \arctan \left (\frac{\sqrt{-x^{2} - 2 \, x + 3}{\left (x + 1\right )}}{x^{2} + 2 \, x - 3}\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 - 1\right ) \left (x + 3\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09121, size = 43, normalized size = 0.83 \begin{align*} \frac{1}{6} \,{\left ({\left (2 \, x + 1\right )} x - 9\right )} \sqrt{-x^{2} - 2 \, x + 3} - 2 \, \arcsin \left (\frac{1}{2} \, x + \frac{1}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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