Optimal. Leaf size=46 \[ -\frac{1}{448} \left (9-4 x^2\right )^{7/2}+\frac{9}{160} \left (9-4 x^2\right )^{5/2}-\frac{27}{64} \left (9-4 x^2\right )^{3/2} \]
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Rubi [A] time = 0.0205247, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ -\frac{1}{448} \left (9-4 x^2\right )^{7/2}+\frac{9}{160} \left (9-4 x^2\right )^{5/2}-\frac{27}{64} \left (9-4 x^2\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^5 \sqrt{9-4 x^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \sqrt{9-4 x} x^2 \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{81}{16} \sqrt{9-4 x}-\frac{9}{8} (9-4 x)^{3/2}+\frac{1}{16} (9-4 x)^{5/2}\right ) \, dx,x,x^2\right )\\ &=-\frac{27}{64} \left (9-4 x^2\right )^{3/2}+\frac{9}{160} \left (9-4 x^2\right )^{5/2}-\frac{1}{448} \left (9-4 x^2\right )^{7/2}\\ \end{align*}
Mathematica [A] time = 0.0120286, size = 27, normalized size = 0.59 \[ -\frac{1}{280} \left (9-4 x^2\right )^{3/2} \left (10 x^4+18 x^2+27\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 34, normalized size = 0.7 \begin{align*}{\frac{ \left ( -3+2\,x \right ) \left ( 3+2\,x \right ) \left ( 10\,{x}^{4}+18\,{x}^{2}+27 \right ) }{280}\sqrt{-4\,{x}^{2}+9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 3.88775, size = 54, normalized size = 1.17 \begin{align*} -\frac{1}{28} \,{\left (-4 \, x^{2} + 9\right )}^{\frac{3}{2}} x^{4} - \frac{9}{140} \,{\left (-4 \, x^{2} + 9\right )}^{\frac{3}{2}} x^{2} - \frac{27}{280} \,{\left (-4 \, x^{2} + 9\right )}^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49467, size = 77, normalized size = 1.67 \begin{align*} \frac{1}{280} \,{\left (40 \, x^{6} - 18 \, x^{4} - 54 \, x^{2} - 243\right )} \sqrt{-4 \, x^{2} + 9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.92776, size = 61, normalized size = 1.33 \begin{align*} \frac{x^{6} \sqrt{9 - 4 x^{2}}}{7} - \frac{9 x^{4} \sqrt{9 - 4 x^{2}}}{140} - \frac{27 x^{2} \sqrt{9 - 4 x^{2}}}{140} - \frac{243 \sqrt{9 - 4 x^{2}}}{280} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.33858, size = 70, normalized size = 1.52 \begin{align*} \frac{1}{448} \,{\left (4 \, x^{2} - 9\right )}^{3} \sqrt{-4 \, x^{2} + 9} + \frac{9}{160} \,{\left (4 \, x^{2} - 9\right )}^{2} \sqrt{-4 \, x^{2} + 9} - \frac{27}{64} \,{\left (-4 \, x^{2} + 9\right )}^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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