Optimal. Leaf size=50 \[ -\frac{1}{3} \left (2 x-x^2\right )^{3/2}-\frac{1}{2} (1-x) \sqrt{2 x-x^2}-\frac{1}{2} \sin ^{-1}(1-x) \]
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Rubi [A] time = 0.0109595, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {640, 612, 619, 216} \[ -\frac{1}{3} \left (2 x-x^2\right )^{3/2}-\frac{1}{2} (1-x) \sqrt{2 x-x^2}-\frac{1}{2} \sin ^{-1}(1-x) \]
Antiderivative was successfully verified.
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Rule 640
Rule 612
Rule 619
Rule 216
Rubi steps
\begin{align*} \int x \sqrt{2 x-x^2} \, dx &=-\frac{1}{3} \left (2 x-x^2\right )^{3/2}+\int \sqrt{2 x-x^2} \, dx\\ &=-\frac{1}{2} (1-x) \sqrt{2 x-x^2}-\frac{1}{3} \left (2 x-x^2\right )^{3/2}+\frac{1}{2} \int \frac{1}{\sqrt{2 x-x^2}} \, dx\\ &=-\frac{1}{2} (1-x) \sqrt{2 x-x^2}-\frac{1}{3} \left (2 x-x^2\right )^{3/2}-\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{4}}} \, dx,x,2-2 x\right )\\ &=-\frac{1}{2} (1-x) \sqrt{2 x-x^2}-\frac{1}{3} \left (2 x-x^2\right )^{3/2}-\frac{1}{2} \sin ^{-1}(1-x)\\ \end{align*}
Mathematica [A] time = 0.0484477, size = 39, normalized size = 0.78 \[ \frac{1}{6} \sqrt{-(x-2) x} \left (2 x^2-x-3\right )-\sin ^{-1}\left (\sqrt{1-\frac{x}{2}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 39, normalized size = 0.8 \begin{align*} -{\frac{1}{3} \left ( -{x}^{2}+2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{2-2\,x}{4}\sqrt{-{x}^{2}+2\,x}}+{\frac{\arcsin \left ( -1+x \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.78255, size = 66, normalized size = 1.32 \begin{align*} -\frac{1}{3} \,{\left (-x^{2} + 2 \, x\right )}^{\frac{3}{2}} + \frac{1}{2} \, \sqrt{-x^{2} + 2 \, x} x - \frac{1}{2} \, \sqrt{-x^{2} + 2 \, x} - \frac{1}{2} \, \arcsin \left (-x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80531, size = 90, normalized size = 1.8 \begin{align*} \frac{1}{6} \,{\left (2 \, x^{2} - x - 3\right )} \sqrt{-x^{2} + 2 \, x} - \arctan \left (\frac{\sqrt{-x^{2} + 2 \, x}}{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 x \sqrt{- x \left (x - 2\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21428, size = 39, normalized size = 0.78 \begin{align*} \frac{1}{6} \,{\left ({\left (2 \, x - 1\right )} x - 3\right )} \sqrt{-x^{2} + 2 \, x} + \frac{1}{2} \, \arcsin \left (x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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