Optimal. Leaf size=60 \[ \frac{\sqrt{\frac{x^2+3}{2 x^2+1}} \left (2 x^2+1\right ) \text{EllipticF}\left (\tan ^{-1}\left (\sqrt{2} x\right ),\frac{5}{6}\right )}{\sqrt{6} \sqrt{2 x^4+7 x^2+3}} \]
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Rubi [A] time = 0.0074505, antiderivative size = 60, 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} \[ \frac{\sqrt{\frac{x^2+3}{2 x^2+1}} \left (2 x^2+1\right ) F\left (\tan ^{-1}\left (\sqrt{2} x\right )|\frac{5}{6}\right )}{\sqrt{6} \sqrt{2 x^4+7 x^2+3}} \]
Antiderivative was successfully verified.
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Rule 1099
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{3+7 x^2+2 x^4}} \, dx &=\frac{\sqrt{\frac{3+x^2}{1+2 x^2}} \left (1+2 x^2\right ) F\left (\tan ^{-1}\left (\sqrt{2} x\right )|\frac{5}{6}\right )}{\sqrt{6} \sqrt{3+7 x^2+2 x^4}}\\ \end{align*}
Mathematica [C] time = 0.0280144, size = 61, normalized size = 1.02 \[ -\frac{i \sqrt{x^2+3} \sqrt{2 x^2+1} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{2} x\right ),\frac{1}{6}\right )}{\sqrt{6} \sqrt{2 x^4+7 x^2+3}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.054, size = 50, normalized size = 0.8 \begin{align*}{-{\frac{i}{3}}\sqrt{3}{\it EllipticF} \left ({\frac{i}{3}}\sqrt{3}x,\sqrt{6} \right ) \sqrt{3\,{x}^{2}+9}\sqrt{2\,{x}^{2}+1}{\frac{1}{\sqrt{2\,{x}^{4}+7\,{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} + 7 \, 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} + 7 \, 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} + 7 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} + 7 \, x^{2} + 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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