Optimal. Leaf size=52 \[ \frac{\left (x^2+1\right ) \sqrt{\frac{2 x^2+3}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{3}\right )}{\sqrt{3} \sqrt{2 x^4+5 x^2+3}} \]
[Out]
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Rubi [A] time = 0.0222564, antiderivative size = 52, 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 \[ \frac{\left (x^2+1\right ) \sqrt{\frac{2 x^2+3}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{3}\right )}{\sqrt{3} \sqrt{2 x^4+5 x^2+3}} \]
Antiderivative was successfully verified.
[In] Int[1/Sqrt[3 + 5*x^2 + 2*x^4],x]
[Out]
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Rubi in Sympy [A] time = 3.63815, size = 49, normalized size = 0.94 \[ \frac{\sqrt{6} \sqrt{\frac{4 x^{2} + 6}{x^{2} + 1}} \left (6 x^{2} + 6\right ) F\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{3}\right )}{36 \sqrt{2 x^{4} + 5 x^{2} + 3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(2*x**4+5*x**2+3)**(1/2),x)
[Out]
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Mathematica [C] time = 0.0421946, size = 58, normalized size = 1.12 \[ -\frac{i \sqrt{x^2+1} \sqrt{2 x^2+3} F\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{3}} x\right )|\frac{3}{2}\right )}{\sqrt{4 x^4+10 x^2+6}} \]
Antiderivative was successfully verified.
[In] Integrate[1/Sqrt[3 + 5*x^2 + 2*x^4],x]
[Out]
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Maple [C] time = 0.049, size = 50, normalized size = 1. \[{-{\frac{i}{6}}\sqrt{6}\sqrt{6\,{x}^{2}+9}\sqrt{{x}^{2}+1}{\it EllipticF} \left ({\frac{i}{3}}x\sqrt{6},{\frac{\sqrt{6}}{2}} \right ){\frac{1}{\sqrt{2\,{x}^{4}+5\,{x}^{2}+3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(2*x^4+5*x^2+3)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{2 \, x^{4} + 5 \, x^{2} + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(2*x^4 + 5*x^2 + 3),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{2 \, x^{4} + 5 \, x^{2} + 3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(2*x^4 + 5*x^2 + 3),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{2 x^{4} + 5 x^{2} + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(2*x**4+5*x**2+3)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{2 \, x^{4} + 5 \, x^{2} + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(2*x^4 + 5*x^2 + 3),x, algorithm="giac")
[Out]