Optimal. Leaf size=24 \[ \frac{x}{\sqrt{3-x^2}}-\sin ^{-1}\left (\frac{x}{\sqrt{3}}\right ) \]
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Rubi [A] time = 0.0053872, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {288, 216} \[ \frac{x}{\sqrt{3-x^2}}-\sin ^{-1}\left (\frac{x}{\sqrt{3}}\right ) \]
Antiderivative was successfully verified.
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Rule 288
Rule 216
Rubi steps
\begin{align*} \int \frac{x^2}{\left (3-x^2\right )^{3/2}} \, dx &=\frac{x}{\sqrt{3-x^2}}-\int \frac{1}{\sqrt{3-x^2}} \, dx\\ &=\frac{x}{\sqrt{3-x^2}}-\sin ^{-1}\left (\frac{x}{\sqrt{3}}\right )\\ \end{align*}
Mathematica [A] time = 0.0156706, size = 24, normalized size = 1. \[ \frac{x}{\sqrt{3-x^2}}-\sin ^{-1}\left (\frac{x}{\sqrt{3}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 22, normalized size = 0.9 \begin{align*} -\arcsin \left ({\frac{x\sqrt{3}}{3}} \right ) +{x{\frac{1}{\sqrt{-{x}^{2}+3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.40437, size = 28, normalized size = 1.17 \begin{align*} \frac{x}{\sqrt{-x^{2} + 3}} - \arcsin \left (\frac{1}{3} \, \sqrt{3} x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.14949, size = 90, normalized size = 3.75 \begin{align*} \frac{{\left (x^{2} - 3\right )} \arctan \left (\frac{\sqrt{-x^{2} + 3}}{x}\right ) - \sqrt{-x^{2} + 3} x}{x^{2} - 3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.516362, size = 49, normalized size = 2.04 \begin{align*} - \frac{x^{2} \operatorname{asin}{\left (\frac{\sqrt{3} x}{3} \right )}}{x^{2} - 3} - \frac{x \sqrt{3 - x^{2}}}{x^{2} - 3} + \frac{3 \operatorname{asin}{\left (\frac{\sqrt{3} x}{3} \right )}}{x^{2} - 3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08977, size = 39, normalized size = 1.62 \begin{align*} -\frac{\sqrt{-x^{2} + 3} x}{x^{2} - 3} - \arcsin \left (\frac{1}{3} \, \sqrt{3} x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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