Optimal. Leaf size=46 \[ -\frac{1}{320} \left (-4 x^2-9\right )^{5/2}-\frac{3}{32} \left (-4 x^2-9\right )^{3/2}-\frac{81}{64} \sqrt{-4 x^2-9} \]
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Rubi [A] time = 0.0194125, 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} \[ -\frac{1}{320} \left (-4 x^2-9\right )^{5/2}-\frac{3}{32} \left (-4 x^2-9\right )^{3/2}-\frac{81}{64} \sqrt{-4 x^2-9} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^5}{\sqrt{-9-4 x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{-9-4 x}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{81}{16 \sqrt{-9-4 x}}+\frac{9}{8} \sqrt{-9-4 x}+\frac{1}{16} (-9-4 x)^{3/2}\right ) \, dx,x,x^2\right )\\ &=-\frac{81}{64} \sqrt{-9-4 x^2}-\frac{3}{32} \left (-9-4 x^2\right )^{3/2}-\frac{1}{320} \left (-9-4 x^2\right )^{5/2}\\ \end{align*}
Mathematica [A] time = 0.0084343, size = 27, normalized size = 0.59 \[ -\frac{1}{40} \sqrt{-4 x^2-9} \left (2 x^4-6 x^2+27\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 24, normalized size = 0.5 \begin{align*} -{\frac{2\,{x}^{4}-6\,{x}^{2}+27}{40}\sqrt{-4\,{x}^{2}-9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 3.95294, size = 54, normalized size = 1.17 \begin{align*} -\frac{1}{20} \, \sqrt{-4 \, x^{2} - 9} x^{4} + \frac{3}{20} \, \sqrt{-4 \, x^{2} - 9} x^{2} - \frac{27}{40} \, \sqrt{-4 \, x^{2} - 9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.29725, size = 61, normalized size = 1.33 \begin{align*} -\frac{1}{40} \,{\left (2 \, x^{4} - 6 \, x^{2} + 27\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 = 1.16989, size = 49, normalized size = 1.07 \begin{align*} - \frac{x^{4} \sqrt{- 4 x^{2} - 9}}{20} + \frac{3 x^{2} \sqrt{- 4 x^{2} - 9}}{20} - \frac{27 \sqrt{- 4 x^{2} - 9}}{40} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 2.09956, size = 46, normalized size = 1. \begin{align*} -\frac{1}{320} i \,{\left (4 \, x^{2} + 9\right )}^{\frac{5}{2}} + \frac{3}{32} i \,{\left (4 \, x^{2} + 9\right )}^{\frac{3}{2}} - \frac{81}{64} i \, \sqrt{4 \, x^{2} + 9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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