Optimal. Leaf size=45 \[ \sqrt{\frac{2}{\sqrt{89}-5}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{4 x}{\sqrt{5+\sqrt{89}}}\right ),\frac{1}{32} \left (-57-5 \sqrt{89}\right )\right ) \]
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Rubi [A] time = 0.0696016, antiderivative size = 45, 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, 419} \[ \sqrt{\frac{2}{\sqrt{89}-5}} F\left (\sin ^{-1}\left (\frac{4 x}{\sqrt{5+\sqrt{89}}}\right )|\frac{1}{32} \left (-57-5 \sqrt{89}\right )\right ) \]
Antiderivative was successfully verified.
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Rule 1095
Rule 419
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{2+5 x^2-8 x^4}} \, dx &=\left (4 \sqrt{2}\right ) \int \frac{1}{\sqrt{5+\sqrt{89}-16 x^2} \sqrt{-5+\sqrt{89}+16 x^2}} \, dx\\ &=\sqrt{\frac{2}{-5+\sqrt{89}}} F\left (\sin ^{-1}\left (\frac{4 x}{\sqrt{5+\sqrt{89}}}\right )|\frac{1}{32} \left (-57-5 \sqrt{89}\right )\right )\\ \end{align*}
Mathematica [C] time = 0.0428417, size = 52, normalized size = 1.16 \[ -i \sqrt{\frac{2}{5+\sqrt{89}}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{4 x}{\sqrt{\sqrt{89}-5}}\right ),\frac{1}{32} \left (5 \sqrt{89}-57\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.229, size = 76, normalized size = 1.7 \begin{align*} 2\,{\frac{\sqrt{1- \left ( -5/4+1/4\,\sqrt{89} \right ){x}^{2}}\sqrt{1- \left ( -5/4-1/4\,\sqrt{89} \right ){x}^{2}}{\it EllipticF} \left ( 1/2\,x\sqrt{-5+\sqrt{89}},5/8\,i+i/8\sqrt{89} \right ) }{\sqrt{-5+\sqrt{89}}\sqrt{-8\,{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{-8 \, 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{-8 \, x^{4} + 5 \, x^{2} + 2}}{8 \, 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{- 8 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{-8 \, x^{4} + 5 \, x^{2} + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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