Optimal. Leaf size=58 \[ \frac{\sqrt{\frac{x^2+2}{2 x^2+1}} \left (2 x^2+1\right ) \text{EllipticF}\left (\tan ^{-1}\left (\sqrt{2} x\right ),\frac{3}{4}\right )}{2 \sqrt{2 x^4+5 x^2+2}} \]
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Rubi [A] time = 0.0071399, antiderivative size = 58, 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 = {1099} \[ \frac{\sqrt{\frac{x^2+2}{2 x^2+1}} \left (2 x^2+1\right ) F\left (\tan ^{-1}\left (\sqrt{2} x\right )|\frac{3}{4}\right )}{2 \sqrt{2 x^4+5 x^2+2}} \]
Antiderivative was successfully verified.
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Rule 1099
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{2+5 x^2+2 x^4}} \, dx &=\frac{\sqrt{\frac{2+x^2}{1+2 x^2}} \left (1+2 x^2\right ) F\left (\tan ^{-1}\left (\sqrt{2} x\right )|\frac{3}{4}\right )}{2 \sqrt{2+5 x^2+2 x^4}}\\ \end{align*}
Mathematica [C] time = 0.0224686, size = 58, normalized size = 1. \[ -\frac{i \sqrt{x^2+2} \sqrt{2 x^2+1} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{2} x\right ),\frac{1}{4}\right )}{2 \sqrt{2 x^4+5 x^2+2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.053, size = 48, normalized size = 0.8 \begin{align*}{-{\frac{i}{2}}\sqrt{2}{\it EllipticF} \left ({\frac{i}{2}}x\sqrt{2},2 \right ) \sqrt{2\,{x}^{2}+4}\sqrt{2\,{x}^{2}+1}{\frac{1}{\sqrt{2\,{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{2 \, 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{2 \, 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{2 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{2 \, x^{4} + 5 \, x^{2} + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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