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 = 0.01, antiderivative size = 50, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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) \end {gather*}
Antiderivative was successfully verified.
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Rule 216
Rule 612
Rule 619
Rule 640
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*}
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Mathematica [A] time = 0.05, size = 39, normalized size = 0.78 \begin {gather*} \frac {1}{6} \sqrt {-((x-2) x)} \left (2 x^2-x-3\right )-\sin ^{-1}\left (\sqrt {1-\frac {x}{2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.11, size = 48, normalized size = 0.96 \begin {gather*} \frac {1}{6} \sqrt {2 x-x^2} \left (2 x^2-x-3\right )-\tan ^{-1}\left (\frac {\sqrt {2 x-x^2}}{x}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 42, normalized size = 0.84 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 29, normalized size = 0.58 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 39, normalized size = 0.78 \begin {gather*} \frac {\arcsin \left (x -1\right )}{2}-\frac {\left (-x^{2}+2 x \right )^{\frac {3}{2}}}{3}-\frac {\left (-2 x +2\right ) \sqrt {-x^{2}+2 x}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.97, size = 49, normalized size = 0.98 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 42, normalized size = 0.84 \begin {gather*} -\frac {\sqrt {2\,x-x^2}\,\left (-8\,x^2+4\,x+12\right )}{24}-\frac {\ln \left (x-1-\sqrt {-x\,\left (x-2\right )}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \sqrt {- x \left (x - 2\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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