Optimal. Leaf size=63 \[ \frac{\sqrt{x^2+1} \sqrt{3 x^2-2} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{5} x}{\sqrt{3 x^2-2}}\right ),\frac{3}{5}\right )}{\sqrt{5} \sqrt{3 x^4+x^2-2}} \]
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Rubi [A] time = 0.0077939, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {1097} \[ \frac{\sqrt{x^2+1} \sqrt{3 x^2-2} F\left (\sin ^{-1}\left (\frac{\sqrt{5} x}{\sqrt{3 x^2-2}}\right )|\frac{3}{5}\right )}{\sqrt{5} \sqrt{3 x^4+x^2-2}} \]
Antiderivative was successfully verified.
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Rule 1097
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-2+x^2+3 x^4}} \, dx &=\frac{\sqrt{1+x^2} \sqrt{-2+3 x^2} F\left (\sin ^{-1}\left (\frac{\sqrt{5} x}{\sqrt{-2+3 x^2}}\right )|\frac{3}{5}\right )}{\sqrt{5} \sqrt{-2+x^2+3 x^4}}\\ \end{align*}
Mathematica [A] time = 0.0298895, size = 48, normalized size = 0.76 \[ \frac{\sqrt{\left (\frac{2}{3}-x^2\right ) \left (x^2+1\right )} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right ),-\frac{2}{3}\right )}{\sqrt{3 x^4+x^2-2}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.046, size = 43, normalized size = 0.7 \begin{align*}{-{\frac{i}{2}}{\it EllipticF} \left ( ix,{\frac{i}{2}}\sqrt{6} \right ) \sqrt{{x}^{2}+1}\sqrt{-6\,{x}^{2}+4}{\frac{1}{\sqrt{3\,{x}^{4}+{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} + 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} + 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} + 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} + x^{2} - 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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