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

Optimal. Leaf size=59 \[ \frac{\tan ^{-1}\left (\frac{2-3 x}{\sqrt{3} \sqrt{-3 x^2+4 x-2}}\right )}{3 \sqrt{3}}-\frac{1}{6} (2-3 x) \sqrt{-3 x^2+4 x-2} \]

[Out]

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

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Rubi [A]  time = 0.0328287, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{\tan ^{-1}\left (\frac{2-3 x}{\sqrt{3} \sqrt{-3 x^2+4 x-2}}\right )}{3 \sqrt{3}}-\frac{1}{6} (2-3 x) \sqrt{-3 x^2+4 x-2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[-2 + 4*x - 3*x^2],x]

[Out]

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

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Rubi in Sympy [A]  time = 2.00817, size = 53, normalized size = 0.9 \[ - \frac{\left (- 6 x + 4\right ) \sqrt{- 3 x^{2} + 4 x - 2}}{12} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (- 6 x + 4\right )}{6 \sqrt{- 3 x^{2} + 4 x - 2}} \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0487513, size = 54, normalized size = 0.92 \[ \frac{1}{6} \sqrt{-3 x^2+4 x-2} (3 x-2)+\frac{\tan ^{-1}\left (\frac{2-3 x}{\sqrt{-9 x^2+12 x-6}}\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

((-2 + 3*x)*Sqrt[-2 + 4*x - 3*x^2])/6 + ArcTan[(2 - 3*x)/Sqrt[-6 + 12*x - 9*x^2]
]/(3*Sqrt[3])

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Maple [A]  time = 0.007, size = 46, normalized size = 0.8 \[ -{\frac{-6\,x+4}{12}\sqrt{-3\,{x}^{2}+4\,x-2}}-{\frac{\sqrt{3}}{9}\arctan \left ({\sqrt{3} \left ( x-{\frac{2}{3}} \right ){\frac{1}{\sqrt{-3\,{x}^{2}+4\,x-2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-3*x^2+4*x-2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.917032, size = 62, normalized size = 1.05 \[ \frac{1}{2} \, \sqrt{-3 \, x^{2} + 4 \, x - 2} x + \frac{1}{9} i \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{2}{\left (3 \, x - 2\right )}\right ) - \frac{1}{3} \, \sqrt{-3 \, x^{2} + 4 \, x - 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(-3*x^2 + 4*x - 2)*x + 1/9*I*sqrt(3)*arcsinh(1/2*sqrt(2)*(3*x - 2)) - 1/
3*sqrt(-3*x^2 + 4*x - 2)

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Fricas [A]  time = 0.220191, size = 115, normalized size = 1.95 \[ \frac{1}{18} \, \sqrt{3}{\left (\sqrt{3} \sqrt{-3 \, x^{2} + 4 \, x - 2}{\left (3 \, x - 2\right )} + i \, \log \left (\frac{2 i \, \sqrt{3} \sqrt{-3 \, x^{2} + 4 \, x - 2} - 6 \, x + 4}{x}\right ) - i \, \log \left (\frac{-2 i \, \sqrt{3} \sqrt{-3 \, x^{2} + 4 \, x - 2} - 6 \, x + 4}{x}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/18*sqrt(3)*(sqrt(3)*sqrt(-3*x^2 + 4*x - 2)*(3*x - 2) + I*log((2*I*sqrt(3)*sqrt
(-3*x^2 + 4*x - 2) - 6*x + 4)/x) - I*log((-2*I*sqrt(3)*sqrt(-3*x^2 + 4*x - 2) -
6*x + 4)/x))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(-3*x**2 + 4*x - 2), x)

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GIAC/XCAS [A]  time = 0.211172, size = 51, normalized size = 0.86 \[ \frac{1}{9} \, \sqrt{3} i \arcsin \left (\frac{1}{2} \, \sqrt{2} i{\left (3 \, x - 2\right )}\right ) + \frac{1}{6} \, \sqrt{-3 \, x^{2} + 4 \, x - 2}{\left (3 \, x - 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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