3.967 \(\int \frac{x \sqrt{-1+3 x^2}}{\sqrt{2-3 x^2}} \, dx\)

Optimal. Leaf size=39 \[ -\frac{1}{6} \sqrt{2-3 x^2} \sqrt{3 x^2-1}-\frac{1}{12} \sin ^{-1}\left (3-6 x^2\right ) \]

[Out]

-(Sqrt[2 - 3*x^2]*Sqrt[-1 + 3*x^2])/6 - ArcSin[3 - 6*x^2]/12

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Rubi [A]  time = 0.0888087, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{1}{6} \sqrt{2-3 x^2} \sqrt{3 x^2-1}-\frac{1}{12} \sin ^{-1}\left (3-6 x^2\right ) \]

Antiderivative was successfully verified.

[In]  Int[(x*Sqrt[-1 + 3*x^2])/Sqrt[2 - 3*x^2],x]

[Out]

-(Sqrt[2 - 3*x^2]*Sqrt[-1 + 3*x^2])/6 - ArcSin[3 - 6*x^2]/12

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Rubi in Sympy [A]  time = 11.111, size = 49, normalized size = 1.26 \[ - \frac{\sqrt{- 3 x^{2} + 2} \sqrt{3 x^{2} - 1}}{6} - \frac{\operatorname{atan}{\left (\frac{- 18 x^{2} + 9}{6 \sqrt{- 9 x^{4} + 9 x^{2} - 2}} \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-3*x**2 + 2)*sqrt(3*x**2 - 1)/6 - atan((-18*x**2 + 9)/(6*sqrt(-9*x**4 + 9*
x**2 - 2)))/12

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Mathematica [A]  time = 0.0302128, size = 37, normalized size = 0.95 \[ \frac{1}{6} \left (-\sin ^{-1}\left (\sqrt{2-3 x^2}\right )-\sqrt{-9 x^4+9 x^2-2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(x*Sqrt[-1 + 3*x^2])/Sqrt[2 - 3*x^2],x]

[Out]

(-Sqrt[-2 + 9*x^2 - 9*x^4] - ArcSin[Sqrt[2 - 3*x^2]])/6

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Maple [A]  time = 0.013, size = 60, normalized size = 1.5 \[{\frac{1}{12}\sqrt{-3\,{x}^{2}+2}\sqrt{3\,{x}^{2}-1} \left ( \arcsin \left ( 6\,{x}^{2}-3 \right ) -2\,\sqrt{-9\,{x}^{4}+9\,{x}^{2}-2} \right ){\frac{1}{\sqrt{-9\,{x}^{4}+9\,{x}^{2}-2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(3*x^2-1)^(1/2)*(-3*x^2+2)^(1/2)*(arcsin(6*x^2-3)-2*(-9*x^4+9*x^2-2)^(1/2))
/(-9*x^4+9*x^2-2)^(1/2)

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Maxima [A]  time = 1.49738, size = 36, normalized size = 0.92 \[ -\frac{1}{6} \, \sqrt{-9 \, x^{4} + 9 \, x^{2} - 2} + \frac{1}{12} \, \arcsin \left (6 \, x^{2} - 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(3*x^2 - 1)*x/sqrt(-3*x^2 + 2),x, algorithm="maxima")

[Out]

-1/6*sqrt(-9*x^4 + 9*x^2 - 2) + 1/12*arcsin(6*x^2 - 3)

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Fricas [A]  time = 0.233216, size = 69, normalized size = 1.77 \[ -\frac{1}{6} \, \sqrt{3 \, x^{2} - 1} \sqrt{-3 \, x^{2} + 2} + \frac{1}{12} \, \arctan \left (\frac{3 \,{\left (2 \, x^{2} - 1\right )}}{2 \, \sqrt{3 \, x^{2} - 1} \sqrt{-3 \, x^{2} + 2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(3*x^2 - 1)*x/sqrt(-3*x^2 + 2),x, algorithm="fricas")

[Out]

-1/6*sqrt(3*x^2 - 1)*sqrt(-3*x^2 + 2) + 1/12*arctan(3/2*(2*x^2 - 1)/(sqrt(3*x^2
- 1)*sqrt(-3*x^2 + 2)))

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Sympy [A]  time = 10.969, size = 66, normalized size = 1.69 \[ \frac{\begin{cases} - \frac{\sqrt{- 3 x^{2} + 2} \sqrt{3 x^{2} - 1}}{2} + \frac{\operatorname{asin}{\left (\sqrt{3 x^{2} - 1} \right )}}{2} & \text{for}\: \left (x \geq \frac{\sqrt{3}}{3} \wedge x < \frac{\sqrt{6}}{3}\right ) \vee \left (x \leq - \frac{\sqrt{3}}{3} \wedge x > - \frac{\sqrt{6}}{3}\right ) \end{cases}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-sqrt(-3*x**2 + 2)*sqrt(3*x**2 - 1)/2 + asin(sqrt(3*x**2 - 1))/2, ((x
>= sqrt(3)/3) & (x < sqrt(6)/3)) | ((x <= -sqrt(3)/3) & (x > -sqrt(6)/3))))/3

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GIAC/XCAS [A]  time = 0.232529, size = 45, normalized size = 1.15 \[ -\frac{1}{6} \, \sqrt{3 \, x^{2} - 1} \sqrt{-3 \, x^{2} + 2} + \frac{1}{6} \, \arcsin \left (\sqrt{3 \, x^{2} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(3*x^2 - 1)*x/sqrt(-3*x^2 + 2),x, algorithm="giac")

[Out]

-1/6*sqrt(3*x^2 - 1)*sqrt(-3*x^2 + 2) + 1/6*arcsin(sqrt(3*x^2 - 1))