Optimal. Leaf size=31 \[ \frac{1}{16} \sqrt{4 x^2+9}+\frac{9}{16 \sqrt{4 x^2+9}} \]
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Rubi [A] time = 0.0146203, 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} \[ \frac{1}{16} \sqrt{4 x^2+9}+\frac{9}{16 \sqrt{4 x^2+9}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^3}{\left (9+4 x^2\right )^{3/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{(9+4 x)^{3/2}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{9}{4 (9+4 x)^{3/2}}+\frac{1}{4 \sqrt{9+4 x}}\right ) \, dx,x,x^2\right )\\ &=\frac{9}{16 \sqrt{9+4 x^2}}+\frac{1}{16} \sqrt{9+4 x^2}\\ \end{align*}
Mathematica [A] time = 0.0076352, size = 22, normalized size = 0.71 \[ \frac{2 x^2+9}{8 \sqrt{4 x^2+9}} \]
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}{8}{\frac{1}{\sqrt{4\,{x}^{2}+9}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.41136, size = 35, normalized size = 1.13 \begin{align*} \frac{x^{2}}{4 \, \sqrt{4 \, x^{2} + 9}} + \frac{9}{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 = 1.82282, size = 45, normalized size = 1.45 \begin{align*} \frac{2 \, x^{2} + 9}{8 \, \sqrt{4 \, x^{2} + 9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.546797, size = 27, normalized size = 0.87 \begin{align*} \frac{x^{2}}{4 \sqrt{4 x^{2} + 9}} + \frac{9}{8 \sqrt{4 x^{2} + 9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.04861, size = 31, normalized size = 1. \begin{align*} \frac{1}{16} \, \sqrt{4 \, x^{2} + 9} + \frac{9}{16 \, \sqrt{4 \, x^{2} + 9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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