Optimal. Leaf size=92 \[ \frac{\left (\sqrt{6} x^2+3\right ) \sqrt{\frac{2 x^4-5 x^2+3}{\left (\sqrt{6} x^2+3\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{2}{3}} x\right ),\frac{1}{24} \left (12+5 \sqrt{6}\right )\right )}{2 \sqrt [4]{6} \sqrt{2 x^4-5 x^2+3}} \]
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Rubi [A] time = 0.0091571, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {1096} \[ \frac{\left (\sqrt{6} x^2+3\right ) \sqrt{\frac{2 x^4-5 x^2+3}{\left (\sqrt{6} x^2+3\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{2}{3}} x\right )|\frac{1}{24} \left (12+5 \sqrt{6}\right )\right )}{2 \sqrt [4]{6} \sqrt{2 x^4-5 x^2+3}} \]
Antiderivative was successfully verified.
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Rule 1096
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{3-5 x^2+2 x^4}} \, dx &=\frac{\left (3+\sqrt{6} x^2\right ) \sqrt{\frac{3-5 x^2+2 x^4}{\left (3+\sqrt{6} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{2}{3}} x\right )|\frac{1}{24} \left (12+5 \sqrt{6}\right )\right )}{2 \sqrt [4]{6} \sqrt{3-5 x^2+2 x^4}}\\ \end{align*}
Mathematica [A] time = 0.0257186, size = 53, normalized size = 0.58 \[ \frac{\sqrt{3-2 x^2} \sqrt{1-x^2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{3}} x\right ),\frac{3}{2}\right )}{\sqrt{4 x^4-10 x^2+6}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 42, normalized size = 0.5 \begin{align*}{\frac{1}{3}\sqrt{-{x}^{2}+1}\sqrt{-6\,{x}^{2}+9}{\it EllipticF} \left ( x,{\frac{\sqrt{6}}{3}} \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{1}{\sqrt{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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