Optimal. Leaf size=18 \[ \frac{\text{EllipticF}\left (\sin ^{-1}\left (\sqrt{2} x\right ),-\frac{1}{6}\right )}{\sqrt{6}} \]
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Rubi [A] time = 0.0122699, antiderivative size = 18, 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, 419} \[ \frac{F\left (\sin ^{-1}\left (\sqrt{2} x\right )|-\frac{1}{6}\right )}{\sqrt{6}} \]
Antiderivative was successfully verified.
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Rule 1095
Rule 419
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{2-4 x^2} \sqrt{12+4 x^2}} \, dx\\ &=\frac{F\left (\sin ^{-1}\left (\sqrt{2} x\right )|-\frac{1}{6}\right )}{\sqrt{6}}\\ \end{align*}
Mathematica [B] time = 0.0243992, size = 54, normalized size = 3. \[ \frac{\sqrt{1-2 x^2} \sqrt{x^2+3} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{2} x\right ),-\frac{1}{6}\right )}{\sqrt{6} \sqrt{-2 x^4-5 x^2+3}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.056, size = 50, normalized size = 2.8 \begin{align*}{\frac{\sqrt{2}{\it EllipticF} \left ( x\sqrt{2},{\frac{i}{6}}\sqrt{6} \right ) }{6}\sqrt{-2\,{x}^{2}+1}\sqrt{3\,{x}^{2}+9}{\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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