Optimal. Leaf size=19 \[ -\frac{E\left (\left .\cos ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.0065772, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {425} \[ -\frac{E\left (\left .\cos ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 425
Rubi steps
\begin{align*} \int \frac{\sqrt{-1+3 x^2}}{\sqrt{2-3 x^2}} \, dx &=-\frac{E\left (\left .\cos ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{\sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0259826, size = 35, normalized size = 1.84 \[ \frac{\sqrt{3 x^2-1} E\left (\left .\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{\sqrt{3-9 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 37, normalized size = 2. \begin{align*} -{\frac{\sqrt{3}}{3}{\it EllipticE} \left ({\frac{x\sqrt{2}\sqrt{3}}{2}},\sqrt{2} \right ) \sqrt{-3\,{x}^{2}+1}{\frac{1}{\sqrt{3\,{x}^{2}-1}}}} \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{3 \, x^{2} - 1}}{\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{3 \, x^{2} - 1} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{3 x^{2} - 1}}{\sqrt{2 - 3 x^{2}}}\, dx \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{3 \, x^{2} - 1}}{\sqrt{-3 \, x^{2} + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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