Optimal. Leaf size=45 \[ \frac{1}{4} \sqrt{9-4 x^2} x^3-\frac{9}{32} \sqrt{9-4 x^2} x+\frac{81}{64} \sin ^{-1}\left (\frac{2 x}{3}\right ) \]
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Rubi [A] time = 0.0104155, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {279, 321, 216} \[ \frac{1}{4} \sqrt{9-4 x^2} x^3-\frac{9}{32} \sqrt{9-4 x^2} x+\frac{81}{64} \sin ^{-1}\left (\frac{2 x}{3}\right ) \]
Antiderivative was successfully verified.
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Rule 279
Rule 321
Rule 216
Rubi steps
\begin{align*} \int x^2 \sqrt{9-4 x^2} \, dx &=\frac{1}{4} x^3 \sqrt{9-4 x^2}+\frac{9}{4} \int \frac{x^2}{\sqrt{9-4 x^2}} \, dx\\ &=-\frac{9}{32} x \sqrt{9-4 x^2}+\frac{1}{4} x^3 \sqrt{9-4 x^2}+\frac{81}{32} \int \frac{1}{\sqrt{9-4 x^2}} \, dx\\ &=-\frac{9}{32} x \sqrt{9-4 x^2}+\frac{1}{4} x^3 \sqrt{9-4 x^2}+\frac{81}{64} \sin ^{-1}\left (\frac{2 x}{3}\right )\\ \end{align*}
Mathematica [A] time = 0.0155067, size = 36, normalized size = 0.8 \[ \sqrt{9-4 x^2} \left (\frac{x^3}{4}-\frac{9 x}{32}\right )+\frac{81}{64} \sin ^{-1}\left (\frac{2 x}{3}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 32, normalized size = 0.7 \begin{align*} -{\frac{x}{16} \left ( -4\,{x}^{2}+9 \right ) ^{{\frac{3}{2}}}}+{\frac{9\,x}{32}\sqrt{-4\,{x}^{2}+9}}+{\frac{81}{64}\arcsin \left ({\frac{2\,x}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 3.07358, size = 42, normalized size = 0.93 \begin{align*} -\frac{1}{16} \,{\left (-4 \, x^{2} + 9\right )}^{\frac{3}{2}} x + \frac{9}{32} \, \sqrt{-4 \, x^{2} + 9} x + \frac{81}{64} \, \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 = 1.54845, size = 111, normalized size = 2.47 \begin{align*} \frac{1}{32} \,{\left (8 \, x^{3} - 9 \, x\right )} \sqrt{-4 \, x^{2} + 9} - \frac{81}{32} \, \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 = 2.65184, size = 124, normalized size = 2.76 \begin{align*} \begin{cases} \frac{i x^{5}}{\sqrt{4 x^{2} - 9}} - \frac{27 i x^{3}}{8 \sqrt{4 x^{2} - 9}} + \frac{81 i x}{32 \sqrt{4 x^{2} - 9}} - \frac{81 i \operatorname{acosh}{\left (\frac{2 x}{3} \right )}}{64} & \text{for}\: \frac{4 \left |{x^{2}}\right |}{9} > 1 \\- \frac{x^{5}}{\sqrt{9 - 4 x^{2}}} + \frac{27 x^{3}}{8 \sqrt{9 - 4 x^{2}}} - \frac{81 x}{32 \sqrt{9 - 4 x^{2}}} + \frac{81 \operatorname{asin}{\left (\frac{2 x}{3} \right )}}{64} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.25919, size = 35, normalized size = 0.78 \begin{align*} \frac{1}{32} \,{\left (8 \, x^{2} - 9\right )} \sqrt{-4 \, x^{2} + 9} x + \frac{81}{64} \, \arcsin \left (\frac{2}{3} \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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