Optimal. Leaf size=35 \[ -\frac{1}{2} \sqrt{6 x-x^2} (3-x)-\frac{9}{2} \sin ^{-1}\left (1-\frac{x}{3}\right ) \]
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Rubi [A] time = 0.0098087, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {612, 619, 216} \[ -\frac{1}{2} \sqrt{6 x-x^2} (3-x)-\frac{9}{2} \sin ^{-1}\left (1-\frac{x}{3}\right ) \]
Antiderivative was successfully verified.
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Rule 612
Rule 619
Rule 216
Rubi steps
\begin{align*} \int \sqrt{6 x-x^2} \, dx &=-\frac{1}{2} (3-x) \sqrt{6 x-x^2}+\frac{9}{2} \int \frac{1}{\sqrt{6 x-x^2}} \, dx\\ &=-\frac{1}{2} (3-x) \sqrt{6 x-x^2}-\frac{3}{4} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{36}}} \, dx,x,6-2 x\right )\\ &=-\frac{1}{2} (3-x) \sqrt{6 x-x^2}-\frac{9}{2} \sin ^{-1}\left (1-\frac{x}{3}\right )\\ \end{align*}
Mathematica [A] time = 0.0427556, size = 32, normalized size = 0.91 \[ \frac{1}{2} (x-3) \sqrt{-(x-6) x}-9 \sin ^{-1}\left (\sqrt{1-\frac{x}{6}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 28, normalized size = 0.8 \begin{align*} -{\frac{-2\,x+6}{4}\sqrt{-{x}^{2}+6\,x}}+{\frac{9}{2}\arcsin \left ( -1+{\frac{x}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.78977, size = 49, normalized size = 1.4 \begin{align*} \frac{1}{2} \, \sqrt{-x^{2} + 6 \, x} x - \frac{3}{2} \, \sqrt{-x^{2} + 6 \, x} - \frac{9}{2} \, \arcsin \left (-\frac{1}{3} \, x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.17013, size = 82, normalized size = 2.34 \begin{align*} \frac{1}{2} \, \sqrt{-x^{2} + 6 \, x}{\left (x - 3\right )} - 9 \, \arctan \left (\frac{\sqrt{-x^{2} + 6 \, 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 \sqrt{- x^{2} + 6 x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28916, size = 34, normalized size = 0.97 \begin{align*} \frac{1}{2} \, \sqrt{-x^{2} + 6 \, x}{\left (x - 3\right )} + \frac{9}{2} \, \arcsin \left (\frac{1}{3} \, x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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