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 = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.00, size = 21, normalized size = 1.00 \[ \frac {2 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|-\frac {1}{6}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {x^{2} + 4} \sqrt {-3 \, x^{2} + 2}}{3 \, x^{2} - 2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {x^{2} + 4}}{\sqrt {-3 \, x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 19, normalized size = 0.90 \[ \frac {2 \sqrt {3}\, \EllipticE \left (\frac {\sqrt {6}\, x}{2}, \frac {i \sqrt {6}}{6}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {x^{2} + 4}}{\sqrt {-3 \, x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {\sqrt {x^2+4}}{\sqrt {2-3\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.17, size = 37, normalized size = 1.76 \[ \begin {cases} \frac {2 \sqrt {3} E\left (\operatorname {asin}{\left (\frac {\sqrt {6} x}{2} \right )}\middle | - \frac {1}{6}\right )}{3} & \text {for}\: x > - \frac {\sqrt {6}}{3} \wedge x < \frac {\sqrt {6}}{3} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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