Optimal. Leaf size=52 \[ \frac{\left (x^2+1\right ) \sqrt{\frac{3 x^2+2}{x^2+1}} \text{EllipticF}\left (\tan ^{-1}(x),-\frac{1}{2}\right )}{\sqrt{2} \sqrt{3 x^4+5 x^2+2}} \]
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Rubi [A] time = 0.007413, antiderivative size = 52, 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 = {1100} \[ \frac{\left (x^2+1\right ) \sqrt{\frac{3 x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|-\frac{1}{2}\right )}{\sqrt{2} \sqrt{3 x^4+5 x^2+2}} \]
Antiderivative was successfully verified.
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Rule 1100
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{2+5 x^2+3 x^4}} \, dx &=\frac{\left (1+x^2\right ) \sqrt{\frac{2+3 x^2}{1+x^2}} F\left (\tan ^{-1}(x)|-\frac{1}{2}\right )}{\sqrt{2} \sqrt{2+5 x^2+3 x^4}}\\ \end{align*}
Mathematica [C] time = 0.0252434, size = 58, normalized size = 1.12 \[ -\frac{i \sqrt{x^2+1} \sqrt{3 x^2+2} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ),\frac{2}{3}\right )}{\sqrt{9 x^4+15 x^2+6}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 44, normalized size = 0.9 \begin{align*}{-{\frac{i}{2}}\sqrt{{x}^{2}+1}\sqrt{6\,{x}^{2}+4}{\it EllipticF} \left ( ix,{\frac{\sqrt{6}}{2}} \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{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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