Optimal. Leaf size=46 \[ -\frac{1}{448} \left (-4 x^2-9\right )^{7/2}-\frac{9}{160} \left (-4 x^2-9\right )^{5/2}-\frac{27}{64} \left (-4 x^2-9\right )^{3/2} \]
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Rubi [A] time = 0.0220785, 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} \[ -\frac{1}{448} \left (-4 x^2-9\right )^{7/2}-\frac{9}{160} \left (-4 x^2-9\right )^{5/2}-\frac{27}{64} \left (-4 x^2-9\right )^{3/2} \]
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 = 0.0123819, size = 27, normalized size = 0.59 \[ -\frac{1}{280} \left (-4 x^2-9\right )^{3/2} \left (10 x^4-18 x^2+27\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 24, normalized size = 0.5 \begin{align*} -{\frac{10\,{x}^{4}-18\,{x}^{2}+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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Maxima [A] time = 3.08869, 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 = 1.27527, 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 = 1.91907, size = 68, normalized size = 1.48 \begin{align*} \frac{x^{6} \sqrt{- 4 x^{2} - 9}}{7} + \frac{9 x^{4} \sqrt{- 4 x^{2} - 9}}{140} - \frac{27 x^{2} \sqrt{- 4 x^{2} - 9}}{140} + \frac{243 \sqrt{- 4 x^{2} - 9}}{280} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 2.18085, size = 46, normalized size = 1. \begin{align*} \frac{1}{448} i \,{\left (4 \, x^{2} + 9\right )}^{\frac{7}{2}} - \frac{9}{160} i \,{\left (4 \, x^{2} + 9\right )}^{\frac{5}{2}} + \frac{27}{64} i \,{\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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