3.176 \(\int \sqrt{3-6 x^2} \sqrt{2+4 x^2} \, dx\)

Optimal. Leaf size=38 \[ \sqrt{\frac{2}{3}} \sqrt{1-4 x^4} x+\frac{2 F\left (\left .\sin ^{-1}\left (\sqrt{2} x\right )\right |-1\right )}{\sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0347745, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ \sqrt{\frac{2}{3}} \sqrt{1-4 x^4} x+\frac{2 F\left (\left .\sin ^{-1}\left (\sqrt{2} x\right )\right |-1\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[3 - 6*x^2]*Sqrt[2 + 4*x^2],x]

[Out]

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

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Rubi in Sympy [A]  time = 5.06177, size = 32, normalized size = 0.84 \[ \frac{x \sqrt{- 24 x^{4} + 6}}{3} + \frac{2 \sqrt{3} F\left (\operatorname{asin}{\left (\sqrt{2} x \right )}\middle | -1\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-6*x**2+3)**(1/2)*(4*x**2+2)**(1/2),x)

[Out]

x*sqrt(-24*x**4 + 6)/3 + 2*sqrt(3)*elliptic_f(asin(sqrt(2)*x), -1)/3

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Mathematica [A]  time = 0.0803951, size = 32, normalized size = 0.84 \[ \frac{\sqrt{2-8 x^4} x+2 F\left (\left .\sin ^{-1}\left (\sqrt{2} x\right )\right |-1\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[3 - 6*x^2]*Sqrt[2 + 4*x^2],x]

[Out]

(x*Sqrt[2 - 8*x^4] + 2*EllipticF[ArcSin[Sqrt[2]*x], -1])/Sqrt[3]

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Maple [B]  time = 0.059, size = 75, normalized size = 2. \[ -{\frac{\sqrt{2}}{36\,{x}^{4}-9}\sqrt{-6\,{x}^{2}+3}\sqrt{2\,{x}^{2}+1} \left ( \sqrt{2}\sqrt{3}\sqrt{-6\,{x}^{2}+3}\sqrt{2\,{x}^{2}+1}{\it EllipticF} \left ( x\sqrt{2},i \right ) -12\,{x}^{5}+3\,x \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-6*x^2+3)^(1/2)*(4*x^2+2)^(1/2),x)

[Out]

-1/9*(-6*x^2+3)^(1/2)*2^(1/2)*(2*x^2+1)^(1/2)*(2^(1/2)*3^(1/2)*(-6*x^2+3)^(1/2)*
(2*x^2+1)^(1/2)*EllipticF(x*2^(1/2),I)-12*x^5+3*x)/(4*x^4-1)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{4 \, x^{2} + 2} \sqrt{-6 \, x^{2} + 3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(4*x^2 + 2)*sqrt(-6*x^2 + 3),x, algorithm="maxima")

[Out]

integrate(sqrt(4*x^2 + 2)*sqrt(-6*x^2 + 3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{4 \, x^{2} + 2} \sqrt{-6 \, x^{2} + 3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(4*x^2 + 2)*sqrt(-6*x^2 + 3),x, algorithm="fricas")

[Out]

integral(sqrt(4*x^2 + 2)*sqrt(-6*x^2 + 3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \sqrt{6} \int \sqrt{- 2 x^{2} + 1} \sqrt{2 x^{2} + 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-6*x**2+3)**(1/2)*(4*x**2+2)**(1/2),x)

[Out]

sqrt(6)*Integral(sqrt(-2*x**2 + 1)*sqrt(2*x**2 + 1), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{4 \, x^{2} + 2} \sqrt{-6 \, x^{2} + 3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(4*x^2 + 2)*sqrt(-6*x^2 + 3),x, algorithm="giac")

[Out]

integrate(sqrt(4*x^2 + 2)*sqrt(-6*x^2 + 3), x)