Optimal. Leaf size=65 \[ \frac{\sqrt{x^2+1} \sqrt{2 x^2-3} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{5} x}{\sqrt{2 x^2-3}}\right ),\frac{2}{5}\right )}{\sqrt{5} \sqrt{2 x^4-x^2-3}} \]
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Rubi [A] time = 0.0068685, antiderivative size = 65, 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} \[ \frac{\sqrt{x^2+1} \sqrt{2 x^2-3} F\left (\sin ^{-1}\left (\frac{\sqrt{5} x}{\sqrt{2 x^2-3}}\right )|\frac{2}{5}\right )}{\sqrt{5} \sqrt{2 x^4-x^2-3}} \]
Antiderivative was successfully verified.
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Rule 1097
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-3-x^2+2 x^4}} \, dx &=\frac{\sqrt{1+x^2} \sqrt{-3+2 x^2} F\left (\sin ^{-1}\left (\frac{\sqrt{5} x}{\sqrt{-3+2 x^2}}\right )|\frac{2}{5}\right )}{\sqrt{5} \sqrt{-3-x^2+2 x^4}}\\ \end{align*}
Mathematica [A] time = 0.0230434, size = 51, normalized size = 0.78 \[ \frac{\sqrt{3-2 x^2} \sqrt{x^2+1} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{3}} x\right ),-\frac{3}{2}\right )}{\sqrt{4 x^4-2 x^2-6}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.052, size = 45, normalized size = 0.7 \begin{align*}{-{\frac{i}{3}}{\it EllipticF} \left ( ix,{\frac{i}{3}}\sqrt{6} \right ) \sqrt{{x}^{2}+1}\sqrt{-6\,{x}^{2}+9}{\frac{1}{\sqrt{2\,{x}^{4}-{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} - 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} - 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} - 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} - x^{2} - 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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