Optimal. Leaf size=63 \[ \frac{\sqrt{x^2-2} \sqrt{3 x^2+1} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{7} x}{\sqrt{x^2-2}}\right ),\frac{1}{7}\right )}{\sqrt{7} \sqrt{3 x^4-5 x^2-2}} \]
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Rubi [A] time = 0.0069716, antiderivative size = 63, 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 = {1097} \[ \frac{\sqrt{x^2-2} \sqrt{3 x^2+1} F\left (\sin ^{-1}\left (\frac{\sqrt{7} x}{\sqrt{x^2-2}}\right )|\frac{1}{7}\right )}{\sqrt{7} \sqrt{3 x^4-5 x^2-2}} \]
Antiderivative was successfully verified.
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Rule 1097
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-2-5 x^2+3 x^4}} \, dx &=\frac{\sqrt{-2+x^2} \sqrt{1+3 x^2} F\left (\sin ^{-1}\left (\frac{\sqrt{7} x}{\sqrt{-2+x^2}}\right )|\frac{1}{7}\right )}{\sqrt{7} \sqrt{-2-5 x^2+3 x^4}}\\ \end{align*}
Mathematica [C] time = 0.0240015, size = 65, normalized size = 1.03 \[ -\frac{i \sqrt{1-\frac{x^2}{2}} \sqrt{3 x^2+1} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{3} x\right ),-\frac{1}{6}\right )}{\sqrt{3} \sqrt{3 x^4-5 x^2-2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.057, size = 53, normalized size = 0.8 \begin{align*}{-{\frac{i}{6}}\sqrt{3}{\it EllipticF} \left ( i\sqrt{3}x,{\frac{i}{6}}\sqrt{6} \right ) \sqrt{3\,{x}^{2}+1}\sqrt{-2\,{x}^{2}+4}{\frac{1}{\sqrt{3\,{x}^{4}-5\,{x}^{2}-2}}}} \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{3 \, x^{4} - 5 \, 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{1}{\sqrt{3 \, x^{4} - 5 \, 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{1}{\sqrt{3 x^{4} - 5 x^{2} - 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{1}{\sqrt{3 \, x^{4} - 5 \, x^{2} - 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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