Optimal. Leaf size=44 \[ -\frac{1}{2} \sqrt{4 x-x^2} x-3 \sqrt{4 x-x^2}-6 \sin ^{-1}\left (1-\frac{x}{2}\right ) \]
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Rubi [A] time = 0.0170626, antiderivative size = 44, 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} \[ -\frac{1}{2} \sqrt{4 x-x^2} x-3 \sqrt{4 x-x^2}-6 \sin ^{-1}\left (1-\frac{x}{2}\right ) \]
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{4 x-x^2}} \, dx &=-\frac{1}{2} x \sqrt{4 x-x^2}+3 \int \frac{x}{\sqrt{4 x-x^2}} \, dx\\ &=-3 \sqrt{4 x-x^2}-\frac{1}{2} x \sqrt{4 x-x^2}+6 \int \frac{1}{\sqrt{4 x-x^2}} \, dx\\ &=-3 \sqrt{4 x-x^2}-\frac{1}{2} x \sqrt{4 x-x^2}-\frac{3}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{16}}} \, dx,x,4-2 x\right )\\ &=-3 \sqrt{4 x-x^2}-\frac{1}{2} x \sqrt{4 x-x^2}-6 \sin ^{-1}\left (1-\frac{x}{2}\right )\\ \end{align*}
Mathematica [A] time = 0.0525605, size = 47, normalized size = 1.07 \[ \frac{1}{2} \left (-\sqrt{4-x} x^{3/2}-6 \sqrt{-(x-4) x}-24 \sin ^{-1}\left (\sqrt{1-\frac{x}{4}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 37, normalized size = 0.8 \begin{align*} 6\,\arcsin \left ( -1+x/2 \right ) -3\,\sqrt{-{x}^{2}+4\,x}-{\frac{x}{2}\sqrt{-{x}^{2}+4\,x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.40573, size = 49, normalized size = 1.11 \begin{align*} -\frac{1}{2} \, \sqrt{-x^{2} + 4 \, x} x - 3 \, \sqrt{-x^{2} + 4 \, x} - 6 \, \arcsin \left (-\frac{1}{2} \, x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.19813, size = 85, normalized size = 1.93 \begin{align*} -\frac{1}{2} \, \sqrt{-x^{2} + 4 \, x}{\left (x + 6\right )} - 12 \, \arctan \left (\frac{\sqrt{-x^{2} + 4 \, 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 - 4\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05737, size = 34, normalized size = 0.77 \begin{align*} -\frac{1}{2} \, \sqrt{-x^{2} + 4 \, x}{\left (x + 6\right )} + 6 \, \arcsin \left (\frac{1}{2} \, x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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