Optimal. Leaf size=21 \[ \frac{2 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|-\frac{1}{6}\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.0064591, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {424} \[ \frac{2 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|-\frac{1}{6}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 424
Rubi steps
\begin{align*} \int \frac{\sqrt{4+x^2}}{\sqrt{2-3 x^2}} \, dx &=\frac{2 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|-\frac{1}{6}\right )}{\sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0041913, size = 21, normalized size = 1. \[ \frac{2 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|-\frac{1}{6}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 19, normalized size = 0.9 \begin{align*}{\frac{2\,\sqrt{3}}{3}{\it EllipticE} \left ({\frac{x\sqrt{6}}{2}},{\frac{i}{6}}\sqrt{6} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x^{2} + 4}}{\sqrt{-3 \, x^{2} + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{x^{2} + 4} \sqrt{-3 \, x^{2} + 2}}{3 \, x^{2} - 2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.51391, size = 37, normalized size = 1.76 \begin{align*} \begin{cases} \frac{2 \sqrt{3} E\left (\operatorname{asin}{\left (\frac{\sqrt{6} x}{2} \right )}\middle | - \frac{1}{6}\right )}{3} & \text{for}\: x > - \frac{\sqrt{6}}{3} \wedge x < \frac{\sqrt{6}}{3} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x^{2} + 4}}{\sqrt{-3 \, x^{2} + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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