Optimal. Leaf size=46 \[ -\frac{1}{2} \sqrt{2 x-x^2} x-\frac{3}{2} \sqrt{2 x-x^2}-\frac{3}{2} \sin ^{-1}(1-x) \]
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Rubi [A] time = 0.0153809, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {670, 640, 619, 216} \[ -\frac{1}{2} \sqrt{2 x-x^2} x-\frac{3}{2} \sqrt{2 x-x^2}-\frac{3}{2} \sin ^{-1}(1-x) \]
Antiderivative was successfully verified.
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Rule 670
Rule 640
Rule 619
Rule 216
Rubi steps
\begin{align*} \int \frac{x^2}{\sqrt{2 x-x^2}} \, dx &=-\frac{1}{2} x \sqrt{2 x-x^2}+\frac{3}{2} \int \frac{x}{\sqrt{2 x-x^2}} \, dx\\ &=-\frac{3}{2} \sqrt{2 x-x^2}-\frac{1}{2} x \sqrt{2 x-x^2}+\frac{3}{2} \int \frac{1}{\sqrt{2 x-x^2}} \, dx\\ &=-\frac{3}{2} \sqrt{2 x-x^2}-\frac{1}{2} x \sqrt{2 x-x^2}-\frac{3}{4} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{4}}} \, dx,x,2-2 x\right )\\ &=-\frac{3}{2} \sqrt{2 x-x^2}-\frac{1}{2} x \sqrt{2 x-x^2}-\frac{3}{2} \sin ^{-1}(1-x)\\ \end{align*}
Mathematica [A] time = 0.0435854, size = 47, normalized size = 1.02 \[ \frac{1}{2} \left (-\sqrt{2-x} x^{3/2}-3 \sqrt{-(x-2) x}-6 \sin ^{-1}\left (\sqrt{1-\frac{x}{2}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 35, normalized size = 0.8 \begin{align*}{\frac{3\,\arcsin \left ( -1+x \right ) }{2}}-{\frac{3}{2}\sqrt{-{x}^{2}+2\,x}}-{\frac{x}{2}\sqrt{-{x}^{2}+2\,x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.73757, size = 49, normalized size = 1.07 \begin{align*} -\frac{1}{2} \, \sqrt{-x^{2} + 2 \, x} x - \frac{3}{2} \, \sqrt{-x^{2} + 2 \, x} - \frac{3}{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 = 1.94772, size = 84, normalized size = 1.83 \begin{align*} -\frac{1}{2} \, \sqrt{-x^{2} + 2 \, x}{\left (x + 3\right )} - 3 \, \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 \frac{x^{2}}{\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 = 1.22651, size = 31, normalized size = 0.67 \begin{align*} -\frac{1}{2} \, \sqrt{-x^{2} + 2 \, x}{\left (x + 3\right )} + \frac{3}{2} \, \arcsin \left (x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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