Optimal. Leaf size=10 \[ \text{EllipticF}\left (\sin ^{-1}\left (\frac{x}{\sqrt{3}}\right ),-6\right ) \]
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Rubi [A] time = 0.0105851, antiderivative size = 10, 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} \[ F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{3}}\right )\right |-6\right ) \]
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{12-4 x^2} \sqrt{2+4 x^2}} \, dx\\ &=F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{3}}\right )\right |-6\right )\\ \end{align*}
Mathematica [C] time = 0.0253669, size = 65, normalized size = 6.5 \[ -\frac{i \sqrt{1-\frac{x^2}{3}} \sqrt{2 x^2+1} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{2} x\right ),-\frac{1}{6}\right )}{\sqrt{2} \sqrt{-2 x^4+5 x^2+3}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.051, size = 51, normalized size = 5.1 \begin{align*}{\frac{\sqrt{3}}{3}\sqrt{-3\,{x}^{2}+9}\sqrt{2\,{x}^{2}+1}{\it EllipticF} \left ({\frac{x\sqrt{3}}{3}},i\sqrt{6} \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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