Optimal. Leaf size=27 \[ \frac{1}{2} \sqrt{9-4 x^2} x+\frac{9}{4} \sin ^{-1}\left (\frac{2 x}{3}\right ) \]
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Rubi [A] time = 0.0032839, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {195, 216} \[ \frac{1}{2} \sqrt{9-4 x^2} x+\frac{9}{4} \sin ^{-1}\left (\frac{2 x}{3}\right ) \]
Antiderivative was successfully verified.
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Rule 195
Rule 216
Rubi steps
\begin{align*} \int \sqrt{9-4 x^2} \, dx &=\frac{1}{2} x \sqrt{9-4 x^2}+\frac{9}{2} \int \frac{1}{\sqrt{9-4 x^2}} \, dx\\ &=\frac{1}{2} x \sqrt{9-4 x^2}+\frac{9}{4} \sin ^{-1}\left (\frac{2 x}{3}\right )\\ \end{align*}
Mathematica [A] time = 0.0059365, size = 27, normalized size = 1. \[ \frac{1}{2} \sqrt{9-4 x^2} x+\frac{9}{4} \sin ^{-1}\left (\frac{2 x}{3}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 20, normalized size = 0.7 \begin{align*}{\frac{9}{4}\arcsin \left ({\frac{2\,x}{3}} \right ) }+{\frac{x}{2}\sqrt{-4\,{x}^{2}+9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 3.62502, size = 26, normalized size = 0.96 \begin{align*} \frac{1}{2} \, \sqrt{-4 \, x^{2} + 9} x + \frac{9}{4} \, \arcsin \left (\frac{2}{3} \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53536, size = 90, normalized size = 3.33 \begin{align*} \frac{1}{2} \, \sqrt{-4 \, x^{2} + 9} x - \frac{9}{2} \, \arctan \left (\frac{\sqrt{-4 \, x^{2} + 9} - 3}{2 \, x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.193277, size = 22, normalized size = 0.81 \begin{align*} \frac{x \sqrt{9 - 4 x^{2}}}{2} + \frac{9 \operatorname{asin}{\left (\frac{2 x}{3} \right )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.56643, size = 26, normalized size = 0.96 \begin{align*} \frac{1}{2} \, \sqrt{-4 \, x^{2} + 9} x + \frac{9}{4} \, \arcsin \left (\frac{2}{3} \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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