Optimal. Leaf size=14 \[ -\text{EllipticF}\left (\cos ^{-1}\left (\sqrt{\frac{2}{3}} x\right ),3\right ) \]
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Rubi [A] time = 0.0116204, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1095, 420} \[ -F\left (\left .\cos ^{-1}\left (\sqrt{\frac{2}{3}} x\right )\right |3\right ) \]
Antiderivative was successfully verified.
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Rule 1095
Rule 420
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-3+5 x^2-2 x^4}} \, dx &=\left (2 \sqrt{2}\right ) \int \frac{1}{\sqrt{6-4 x^2} \sqrt{-4+4 x^2}} \, dx\\ &=-F\left (\left .\cos ^{-1}\left (\sqrt{\frac{2}{3}} x\right )\right |3\right )\\ \end{align*}
Mathematica [B] time = 0.0246362, size = 53, normalized size = 3.79 \[ \frac{\sqrt{3-2 x^2} \sqrt{1-x^2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{3}} x\right ),\frac{3}{2}\right )}{\sqrt{-4 x^4+10 x^2-6}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 50, normalized size = 3.6 \begin{align*}{\frac{\sqrt{6}}{6}\sqrt{-6\,{x}^{2}+9}\sqrt{-{x}^{2}+1}{\it EllipticF} \left ({\frac{x\sqrt{6}}{3}},{\frac{\sqrt{6}}{2}} \right ){\frac{1}{\sqrt{-2\,{x}^{4}+5\,{x}^{2}-3}}}} \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{1}{\sqrt{-2 \, x^{4} + 5 \, x^{2} - 3}}\,{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{-2 \, x^{4} + 5 \, x^{2} - 3}}{2 \, x^{4} - 5 \, x^{2} + 3}, 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{1}{\sqrt{- 2 x^{4} + 5 x^{2} - 3}}\, 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{1}{\sqrt{-2 \, x^{4} + 5 \, x^{2} - 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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