Optimal. Leaf size=63 \[ \frac{1}{6} \sqrt{9-4 x^2} x^5-\frac{3}{32} \sqrt{9-4 x^2} x^3-\frac{81}{256} \sqrt{9-4 x^2} x+\frac{729}{512} \sin ^{-1}\left (\frac{2 x}{3}\right ) \]
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Rubi [A] time = 0.0161792, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {279, 321, 216} \[ \frac{1}{6} \sqrt{9-4 x^2} x^5-\frac{3}{32} \sqrt{9-4 x^2} x^3-\frac{81}{256} \sqrt{9-4 x^2} x+\frac{729}{512} \sin ^{-1}\left (\frac{2 x}{3}\right ) \]
Antiderivative was successfully verified.
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Rule 279
Rule 321
Rule 216
Rubi steps
\begin{align*} \int x^4 \sqrt{9-4 x^2} \, dx &=\frac{1}{6} x^5 \sqrt{9-4 x^2}+\frac{3}{2} \int \frac{x^4}{\sqrt{9-4 x^2}} \, dx\\ &=-\frac{3}{32} x^3 \sqrt{9-4 x^2}+\frac{1}{6} x^5 \sqrt{9-4 x^2}+\frac{81}{32} \int \frac{x^2}{\sqrt{9-4 x^2}} \, dx\\ &=-\frac{81}{256} x \sqrt{9-4 x^2}-\frac{3}{32} x^3 \sqrt{9-4 x^2}+\frac{1}{6} x^5 \sqrt{9-4 x^2}+\frac{729}{256} \int \frac{1}{\sqrt{9-4 x^2}} \, dx\\ &=-\frac{81}{256} x \sqrt{9-4 x^2}-\frac{3}{32} x^3 \sqrt{9-4 x^2}+\frac{1}{6} x^5 \sqrt{9-4 x^2}+\frac{729}{512} \sin ^{-1}\left (\frac{2 x}{3}\right )\\ \end{align*}
Mathematica [A] time = 0.0117474, size = 39, normalized size = 0.62 \[ \frac{1}{768} x \sqrt{9-4 x^2} \left (128 x^4-72 x^2-243\right )+\frac{729}{512} \sin ^{-1}\left (\frac{2 x}{3}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 46, normalized size = 0.7 \begin{align*} -{\frac{{x}^{3}}{24} \left ( -4\,{x}^{2}+9 \right ) ^{{\frac{3}{2}}}}-{\frac{9\,x}{128} \left ( -4\,{x}^{2}+9 \right ) ^{{\frac{3}{2}}}}+{\frac{81\,x}{256}\sqrt{-4\,{x}^{2}+9}}+{\frac{729}{512}\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 = 4.07798, size = 61, normalized size = 0.97 \begin{align*} -\frac{1}{24} \,{\left (-4 \, x^{2} + 9\right )}^{\frac{3}{2}} x^{3} - \frac{9}{128} \,{\left (-4 \, x^{2} + 9\right )}^{\frac{3}{2}} x + \frac{81}{256} \, \sqrt{-4 \, x^{2} + 9} x + \frac{729}{512} \, \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 = 1.54162, size = 132, normalized size = 2.1 \begin{align*} \frac{1}{768} \,{\left (128 \, x^{5} - 72 \, x^{3} - 243 \, x\right )} \sqrt{-4 \, x^{2} + 9} - \frac{729}{256} \, \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 = 4.51964, size = 167, normalized size = 2.65 \begin{align*} \begin{cases} \frac{2 i x^{7}}{3 \sqrt{4 x^{2} - 9}} - \frac{15 i x^{5}}{8 \sqrt{4 x^{2} - 9}} - \frac{27 i x^{3}}{64 \sqrt{4 x^{2} - 9}} + \frac{729 i x}{256 \sqrt{4 x^{2} - 9}} - \frac{729 i \operatorname{acosh}{\left (\frac{2 x}{3} \right )}}{512} & \text{for}\: \frac{4 \left |{x^{2}}\right |}{9} > 1 \\- \frac{2 x^{7}}{3 \sqrt{9 - 4 x^{2}}} + \frac{15 x^{5}}{8 \sqrt{9 - 4 x^{2}}} + \frac{27 x^{3}}{64 \sqrt{9 - 4 x^{2}}} - \frac{729 x}{256 \sqrt{9 - 4 x^{2}}} + \frac{729 \operatorname{asin}{\left (\frac{2 x}{3} \right )}}{512} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.5577, size = 45, normalized size = 0.71 \begin{align*} \frac{1}{768} \,{\left (8 \,{\left (16 \, x^{2} - 9\right )} x^{2} - 243\right )} \sqrt{-4 \, x^{2} + 9} x + \frac{729}{512} \, \arcsin \left (\frac{2}{3} \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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