3.2405 \(\int \frac{x}{\sqrt{-2+4 x-3 x^2}} \, dx\)

Optimal. Leaf size=54 \[ -\frac{1}{3} \sqrt{-3 x^2+4 x-2}-\frac{2 \tan ^{-1}\left (\frac{2-3 x}{\sqrt{3} \sqrt{-3 x^2+4 x-2}}\right )}{3 \sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0456785, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ -\frac{1}{3} \sqrt{-3 x^2+4 x-2}-\frac{2 \tan ^{-1}\left (\frac{2-3 x}{\sqrt{3} \sqrt{-3 x^2+4 x-2}}\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[x/Sqrt[-2 + 4*x - 3*x^2],x]

[Out]

-Sqrt[-2 + 4*x - 3*x^2]/3 - (2*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 = 4.62918, size = 51, normalized size = 0.94 \[ - \frac{\sqrt{- 3 x^{2} + 4 x - 2}}{3} - \frac{2 \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (- 6 x + 4\right )}{6 \sqrt{- 3 x^{2} + 4 x - 2}} \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0507397, size = 49, normalized size = 0.91 \[ \frac{1}{9} \left (-3 \sqrt{-3 x^2+4 x-2}-2 \sqrt{3} \tan ^{-1}\left (\frac{2-3 x}{\sqrt{-9 x^2+12 x-6}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 41, normalized size = 0.8 \[ -{\frac{1}{3}\sqrt{-3\,{x}^{2}+4\,x-2}}+{\frac{2\,\sqrt{3}}{9}\arctan \left ({\sqrt{3} \left ( x-{\frac{2}{3}} \right ){\frac{1}{\sqrt{-3\,{x}^{2}+4\,x-2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(-3*x^2+4*x-2)^(1/2),x)

[Out]

-1/3*(-3*x^2+4*x-2)^(1/2)+2/9*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 = 0.75924, size = 42, normalized size = 0.78 \[ -\frac{2}{9} i \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{2}{\left (3 \, x - 2\right )}\right ) - \frac{1}{3} \, \sqrt{-3 \, x^{2} + 4 \, x - 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/9*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 = 0.221977, size = 108, normalized size = 2. \[ -\frac{1}{9} \, \sqrt{3}{\left (\sqrt{3} \sqrt{-3 \, x^{2} + 4 \, x - 2} + i \, \log \left (\frac{2 i \, \sqrt{3} \sqrt{-3 \, x^{2} + 4 \, x - 2} - 6 \, x + 4}{x}\right ) - i \, \log \left (\frac{-2 i \, \sqrt{3} \sqrt{-3 \, x^{2} + 4 \, x - 2} - 6 \, x + 4}{x}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9*sqrt(3)*(sqrt(3)*sqrt(-3*x^2 + 4*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. \[ \int \frac{x}{\sqrt{- 3 x^{2} + 4 x - 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x/sqrt(-3*x**2 + 4*x - 2), x)

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GIAC/XCAS [A]  time = 0.214733, size = 45, normalized size = 0.83 \[ -\frac{2}{9} \, \sqrt{3} i \arcsin \left (\frac{1}{2} \, \sqrt{2} i{\left (3 \, x - 2\right )}\right ) - \frac{1}{3} \, \sqrt{-3 \, x^{2} + 4 \, x - 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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