Optimal. Leaf size=31 \[ \frac{1}{48} \left (4 x^2+9\right )^{3/2}-\frac{9}{16} \sqrt{4 x^2+9} \]
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Rubi [A] time = 0.0142129, 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} \[ \frac{1}{48} \left (4 x^2+9\right )^{3/2}-\frac{9}{16} \sqrt{4 x^2+9} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^3}{\sqrt{9+4 x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{\sqrt{9+4 x}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{9}{4 \sqrt{9+4 x}}+\frac{1}{4} \sqrt{9+4 x}\right ) \, dx,x,x^2\right )\\ &=-\frac{9}{16} \sqrt{9+4 x^2}+\frac{1}{48} \left (9+4 x^2\right )^{3/2}\\ \end{align*}
Mathematica [A] time = 0.0055979, size = 22, normalized size = 0.71 \[ \frac{1}{24} \left (2 x^2-9\right ) \sqrt{4 x^2+9} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 19, normalized size = 0.6 \begin{align*}{\frac{2\,{x}^{2}-9}{24}\sqrt{4\,{x}^{2}+9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.96505, size = 35, normalized size = 1.13 \begin{align*} \frac{1}{12} \, \sqrt{4 \, x^{2} + 9} x^{2} - \frac{3}{8} \, \sqrt{4 \, x^{2} + 9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.29076, size = 46, normalized size = 1.48 \begin{align*} \frac{1}{24} \, \sqrt{4 \, x^{2} + 9}{\left (2 \, x^{2} - 9\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.339444, size = 27, normalized size = 0.87 \begin{align*} \frac{x^{2} \sqrt{4 x^{2} + 9}}{12} - \frac{3 \sqrt{4 x^{2} + 9}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.1769, size = 31, normalized size = 1. \begin{align*} \frac{1}{48} \,{\left (4 \, x^{2} + 9\right )}^{\frac{3}{2}} - \frac{9}{16} \, \sqrt{4 \, x^{2} + 9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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