Optimal. Leaf size=92 \[ \frac{\left (\sqrt{10} x^2+2\right ) \sqrt{\frac{5 x^4+5 x^2+2}{\left (\sqrt{10} x^2+2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{5}{2}} x\right ),\frac{1}{8} \left (4-\sqrt{10}\right )\right )}{2 \sqrt [4]{10} \sqrt{5 x^4+5 x^2+2}} \]
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Rubi [A] time = 0.0258335, antiderivative size = 92, 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 = {1103} \[ \frac{\left (\sqrt{10} x^2+2\right ) \sqrt{\frac{5 x^4+5 x^2+2}{\left (\sqrt{10} x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{5}{2}} x\right )|\frac{1}{8} \left (4-\sqrt{10}\right )\right )}{2 \sqrt [4]{10} \sqrt{5 x^4+5 x^2+2}} \]
Antiderivative was successfully verified.
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Rule 1103
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{2+5 x^2+5 x^4}} \, dx &=\frac{\left (2+\sqrt{10} x^2\right ) \sqrt{\frac{2+5 x^2+5 x^4}{\left (2+\sqrt{10} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{5}{2}} x\right )|\frac{1}{8} \left (4-\sqrt{10}\right )\right )}{2 \sqrt [4]{10} \sqrt{2+5 x^2+5 x^4}}\\ \end{align*}
Mathematica [C] time = 0.114411, size = 144, normalized size = 1.57 \[ -\frac{i \sqrt{1-\frac{10 x^2}{-5-i \sqrt{15}}} \sqrt{1-\frac{10 x^2}{-5+i \sqrt{15}}} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-\frac{10}{-5-i \sqrt{15}}} x\right ),\frac{-5-i \sqrt{15}}{-5+i \sqrt{15}}\right )}{\sqrt{10} \sqrt{-\frac{1}{-5-i \sqrt{15}}} \sqrt{5 x^4+5 x^2+2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.749, size = 87, normalized size = 1. \begin{align*} 2\,{\frac{\sqrt{1- \left ( -5/4+i/4\sqrt{15} \right ){x}^{2}}\sqrt{1- \left ( -5/4-i/4\sqrt{15} \right ){x}^{2}}{\it EllipticF} \left ( 1/2\,x\sqrt{-5+i\sqrt{15}},1/2\,\sqrt{1+i\sqrt{15}} \right ) }{\sqrt{-5+i\sqrt{15}}\sqrt{5\,{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{5 \, 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{5 \, 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{5 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{5 \, x^{4} + 5 \, x^{2} + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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