Optimal. Leaf size=52 \[ -\frac{\sqrt{-3 x^2-2} \text{EllipticF}\left (\tan ^{-1}(x),-\frac{1}{2}\right )}{\sqrt{2} \sqrt{x^2+1} \sqrt{\frac{3 x^2+2}{x^2+1}}} \]
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Rubi [A] time = 0.0167469, antiderivative size = 52, 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, 418} \[ -\frac{\sqrt{-3 x^2-2} F\left (\tan ^{-1}(x)|-\frac{1}{2}\right )}{\sqrt{2} \sqrt{x^2+1} \sqrt{\frac{3 x^2+2}{x^2+1}}} \]
Antiderivative was successfully verified.
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Rule 1095
Rule 418
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-2-5 x^2-3 x^4}} \, dx &=\left (2 \sqrt{3}\right ) \int \frac{1}{\sqrt{-4-6 x^2} \sqrt{6+6 x^2}} \, dx\\ &=-\frac{\sqrt{-2-3 x^2} F\left (\tan ^{-1}(x)|-\frac{1}{2}\right )}{\sqrt{2} \sqrt{1+x^2} \sqrt{\frac{2+3 x^2}{1+x^2}}}\\ \end{align*}
Mathematica [C] time = 0.024037, size = 63, normalized size = 1.21 \[ -\frac{i \sqrt{x^2+1} \sqrt{3 x^2+2} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ),\frac{2}{3}\right )}{\sqrt{3} \sqrt{-3 x^4-5 x^2-2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.054, size = 50, normalized size = 1. \begin{align*}{-{\frac{i}{6}}\sqrt{6}\sqrt{6\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticF} \left ({\frac{i}{2}}x\sqrt{6},{\frac{\sqrt{6}}{3}} \right ){\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{\sqrt{-3 \, x^{4} - 5 \, x^{2} - 2}}{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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