3.954 \(\int \sqrt{3-2 x^2-x^4} \, dx\)

Optimal. Leaf size=48 \[ \frac{1}{3} \sqrt{-x^4-2 x^2+3} x+\frac{4 F\left (\sin ^{-1}(x)|-\frac{1}{3}\right )}{\sqrt{3}}-\frac{2 E\left (\sin ^{-1}(x)|-\frac{1}{3}\right )}{\sqrt{3}} \]

[Out]

(x*Sqrt[3 - 2*x^2 - x^4])/3 - (2*EllipticE[ArcSin[x], -1/3])/Sqrt[3] + (4*Ellipt
icF[ArcSin[x], -1/3])/Sqrt[3]

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Rubi [A]  time = 0.139859, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312 \[ \frac{1}{3} \sqrt{-x^4-2 x^2+3} x+\frac{4 F\left (\sin ^{-1}(x)|-\frac{1}{3}\right )}{\sqrt{3}}-\frac{2 E\left (\sin ^{-1}(x)|-\frac{1}{3}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[3 - 2*x^2 - x^4],x]

[Out]

(x*Sqrt[3 - 2*x^2 - x^4])/3 - (2*EllipticE[ArcSin[x], -1/3])/Sqrt[3] + (4*Ellipt
icF[ArcSin[x], -1/3])/Sqrt[3]

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Rubi in Sympy [A]  time = 22.9189, size = 49, normalized size = 1.02 \[ \frac{x \sqrt{- x^{4} - 2 x^{2} + 3}}{3} - \frac{2 \sqrt{3} E\left (\operatorname{asin}{\left (x \right )}\middle | - \frac{1}{3}\right )}{3} + \frac{4 \sqrt{3} F\left (\operatorname{asin}{\left (x \right )}\middle | - \frac{1}{3}\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-x**4-2*x**2+3)**(1/2),x)

[Out]

x*sqrt(-x**4 - 2*x**2 + 3)/3 - 2*sqrt(3)*elliptic_e(asin(x), -1/3)/3 + 4*sqrt(3)
*elliptic_f(asin(x), -1/3)/3

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Mathematica [C]  time = 0.10745, size = 59, normalized size = 1.23 \[ \frac{1}{3} \left (\sqrt{-x^4-2 x^2+3} x-4 i F\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{3}}\right )\right |-3\right )-2 i E\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{3}}\right )\right |-3\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[3 - 2*x^2 - x^4],x]

[Out]

(x*Sqrt[3 - 2*x^2 - x^4] - (2*I)*EllipticE[I*ArcSinh[x/Sqrt[3]], -3] - (4*I)*Ell
ipticF[I*ArcSinh[x/Sqrt[3]], -3])/3

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Maple [B]  time = 0.019, size = 114, normalized size = 2.4 \[{\frac{x}{3}\sqrt{-{x}^{4}-2\,{x}^{2}+3}}+{\frac{2\,{\it EllipticF} \left ( x,i/3\sqrt{3} \right ) }{3}\sqrt{-{x}^{2}+1}\sqrt{3\,{x}^{2}+9}{\frac{1}{\sqrt{-{x}^{4}-2\,{x}^{2}+3}}}}+{\frac{2\,{\it EllipticF} \left ( x,i/3\sqrt{3} \right ) -2\,{\it EllipticE} \left ( x,i/3\sqrt{3} \right ) }{3}\sqrt{-{x}^{2}+1}\sqrt{3\,{x}^{2}+9}{\frac{1}{\sqrt{-{x}^{4}-2\,{x}^{2}+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^4-2*x^2+3)^(1/2),x)

[Out]

1/3*x*(-x^4-2*x^2+3)^(1/2)+2/3*(-x^2+1)^(1/2)*(3*x^2+9)^(1/2)/(-x^4-2*x^2+3)^(1/
2)*EllipticF(x,1/3*I*3^(1/2))+2/3*(-x^2+1)^(1/2)*(3*x^2+9)^(1/2)/(-x^4-2*x^2+3)^
(1/2)*(EllipticF(x,1/3*I*3^(1/2))-EllipticE(x,1/3*I*3^(1/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{-x^{4} - 2 \, x^{2} + 3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^4 - 2*x^2 + 3),x, algorithm="maxima")

[Out]

integrate(sqrt(-x^4 - 2*x^2 + 3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{-x^{4} - 2 \, x^{2} + 3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^4 - 2*x^2 + 3),x, algorithm="fricas")

[Out]

integral(sqrt(-x^4 - 2*x^2 + 3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{- x^{4} - 2 x^{2} + 3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x**4-2*x**2+3)**(1/2),x)

[Out]

Integral(sqrt(-x**4 - 2*x**2 + 3), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{-x^{4} - 2 \, x^{2} + 3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^4 - 2*x^2 + 3),x, algorithm="giac")

[Out]

integrate(sqrt(-x^4 - 2*x^2 + 3), x)