Optimal. Leaf size=27 \[ \frac{9}{16} \sin ^{-1}\left (\frac{2 x}{3}\right )-\frac{1}{8} x \sqrt{9-4 x^2} \]
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Rubi [A] time = 0.0050559, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {321, 216} \[ \frac{9}{16} \sin ^{-1}\left (\frac{2 x}{3}\right )-\frac{1}{8} x \sqrt{9-4 x^2} \]
Antiderivative was successfully verified.
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Rule 321
Rule 216
Rubi steps
\begin{align*} \int \frac{x^2}{\sqrt{9-4 x^2}} \, dx &=-\frac{1}{8} x \sqrt{9-4 x^2}+\frac{9}{8} \int \frac{1}{\sqrt{9-4 x^2}} \, dx\\ &=-\frac{1}{8} x \sqrt{9-4 x^2}+\frac{9}{16} \sin ^{-1}\left (\frac{2 x}{3}\right )\\ \end{align*}
Mathematica [A] time = 0.0052729, size = 27, normalized size = 1. \[ \frac{9}{16} \sin ^{-1}\left (\frac{2 x}{3}\right )-\frac{1}{8} x \sqrt{9-4 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 20, normalized size = 0.7 \begin{align*}{\frac{9}{16}\arcsin \left ({\frac{2\,x}{3}} \right ) }-{\frac{x}{8}\sqrt{-4\,{x}^{2}+9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.57791, size = 26, normalized size = 0.96 \begin{align*} -\frac{1}{8} \, \sqrt{-4 \, x^{2} + 9} x + \frac{9}{16} \, \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 = 1.2887, size = 92, normalized size = 3.41 \begin{align*} -\frac{1}{8} \, \sqrt{-4 \, x^{2} + 9} x - \frac{9}{8} \, \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 = 0.202437, size = 22, normalized size = 0.81 \begin{align*} - \frac{x \sqrt{9 - 4 x^{2}}}{8} + \frac{9 \operatorname{asin}{\left (\frac{2 x}{3} \right )}}{16} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.96208, size = 26, normalized size = 0.96 \begin{align*} -\frac{1}{8} \, \sqrt{-4 \, x^{2} + 9} x + \frac{9}{16} \, \arcsin \left (\frac{2}{3} \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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