Optimal. Leaf size=90 \[ \frac{\left (\sqrt{2} x^2+1\right ) \sqrt{\frac{4 x^4+5 x^2+2}{\left (\sqrt{2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{16} \left (8-5 \sqrt{2}\right )\right )}{2\ 2^{3/4} \sqrt{4 x^4+5 x^2+2}} \]
[Out]
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Rubi [A] time = 0.0527348, antiderivative size = 90, 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 \[ \frac{\left (\sqrt{2} x^2+1\right ) \sqrt{\frac{4 x^4+5 x^2+2}{\left (\sqrt{2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{16} \left (8-5 \sqrt{2}\right )\right )}{2\ 2^{3/4} \sqrt{4 x^4+5 x^2+2}} \]
Antiderivative was successfully verified.
[In] Int[1/Sqrt[2 + 5*x^2 + 4*x^4],x]
[Out]
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Rubi in Sympy [A] time = 3.74721, size = 80, normalized size = 0.89 \[ \frac{\sqrt [4]{2} \sqrt{\frac{4 x^{4} + 5 x^{2} + 2}{\left (\sqrt{2} x^{2} + 1\right )^{2}}} \left (\sqrt{2} x^{2} + 1\right ) F\left (2 \operatorname{atan}{\left (\sqrt [4]{2} x \right )}\middle | - \frac{5 \sqrt{2}}{16} + \frac{1}{2}\right )}{4 \sqrt{4 x^{4} + 5 x^{2} + 2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(4*x**4+5*x**2+2)**(1/2),x)
[Out]
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Mathematica [C] time = 0.154818, size = 147, normalized size = 1.63 \[ -\frac{i \sqrt{1-\frac{8 x^2}{-5-i \sqrt{7}}} \sqrt{1-\frac{8 x^2}{-5+i \sqrt{7}}} F\left (i \sinh ^{-1}\left (2 \sqrt{-\frac{2}{-5-i \sqrt{7}}} x\right )|\frac{-5-i \sqrt{7}}{-5+i \sqrt{7}}\right )}{2 \sqrt{2} \sqrt{-\frac{1}{-5-i \sqrt{7}}} \sqrt{4 x^4+5 x^2+2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/Sqrt[2 + 5*x^2 + 4*x^4],x]
[Out]
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Maple [C] time = 0.161, size = 87, normalized size = 1. \[ 2\,{\frac{\sqrt{1- \left ( -5/4+i/4\sqrt{7} \right ){x}^{2}}\sqrt{1- \left ( -5/4-i/4\sqrt{7} \right ){x}^{2}}{\it EllipticF} \left ( 1/2\,x\sqrt{-5+i\sqrt{7}},1/4\,\sqrt{9+5\,i\sqrt{7}} \right ) }{\sqrt{-5+i\sqrt{7}}\sqrt{4\,{x}^{4}+5\,{x}^{2}+2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(4*x^4+5*x^2+2)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{4 \, x^{4} + 5 \, x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(4*x^4 + 5*x^2 + 2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{4 \, x^{4} + 5 \, x^{2} + 2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(4*x^4 + 5*x^2 + 2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{4 x^{4} + 5 x^{2} + 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(4*x**4+5*x**2+2)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{4 \, x^{4} + 5 \, x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(4*x^4 + 5*x^2 + 2),x, algorithm="giac")
[Out]