Optimal. Leaf size=31 \[ \frac{1}{80} \left (-4 x^2-9\right )^{5/2}+\frac{3}{16} \left (-4 x^2-9\right )^{3/2} \]
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Rubi [A] time = 0.0139497, antiderivative size = 31, 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} \[ \frac{1}{80} \left (-4 x^2-9\right )^{5/2}+\frac{3}{16} \left (-4 x^2-9\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^3 \sqrt{-9-4 x^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \sqrt{-9-4 x} x \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{9}{4} \sqrt{-9-4 x}-\frac{1}{4} (-9-4 x)^{3/2}\right ) \, dx,x,x^2\right )\\ &=\frac{3}{16} \left (-9-4 x^2\right )^{3/2}+\frac{1}{80} \left (-9-4 x^2\right )^{5/2}\\ \end{align*}
Mathematica [A] time = 0.0095639, size = 22, normalized size = 0.71 \[ \frac{1}{40} \left (-4 x^2-9\right )^{3/2} \left (3-2 x^2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 19, normalized size = 0.6 \begin{align*} -{\frac{2\,{x}^{2}-3}{40} \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 = 4.20774, size = 35, normalized size = 1.13 \begin{align*} -\frac{1}{20} \,{\left (-4 \, x^{2} - 9\right )}^{\frac{3}{2}} x^{2} + \frac{3}{40} \,{\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 = 1.27106, size = 59, normalized size = 1.9 \begin{align*} \frac{1}{40} \,{\left (8 \, 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 = 0.603163, size = 49, normalized size = 1.58 \begin{align*} \frac{x^{4} \sqrt{- 4 x^{2} - 9}}{5} + \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 = 2.76936, size = 31, normalized size = 1. \begin{align*} \frac{1}{80} i \,{\left (4 \, x^{2} + 9\right )}^{\frac{5}{2}} - \frac{3}{16} 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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