3.113 \(\int \sqrt{-2+5 x-3 x^2} \, dx\)

Optimal. Leaf size=39 \[ -\frac{1}{12} \sqrt{-3 x^2+5 x-2} (5-6 x)-\frac{\sin ^{-1}(5-6 x)}{24 \sqrt{3}} \]

[Out]

-((5 - 6*x)*Sqrt[-2 + 5*x - 3*x^2])/12 - ArcSin[5 - 6*x]/(24*Sqrt[3])

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Rubi [A]  time = 0.0222279, antiderivative size = 39, 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 \[ -\frac{1}{12} \sqrt{-3 x^2+5 x-2} (5-6 x)-\frac{\sin ^{-1}(5-6 x)}{24 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[-2 + 5*x - 3*x^2],x]

[Out]

-((5 - 6*x)*Sqrt[-2 + 5*x - 3*x^2])/12 - ArcSin[5 - 6*x]/(24*Sqrt[3])

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Rubi in Sympy [A]  time = 2.06051, size = 54, normalized size = 1.38 \[ - \frac{\left (- 6 x + 5\right ) \sqrt{- 3 x^{2} + 5 x - 2}}{12} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (- 6 x + 5\right )}{6 \sqrt{- 3 x^{2} + 5 x - 2}} \right )}}{72} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-3*x**2+5*x-2)**(1/2),x)

[Out]

-(-6*x + 5)*sqrt(-3*x**2 + 5*x - 2)/12 - sqrt(3)*atan(sqrt(3)*(-6*x + 5)/(6*sqrt
(-3*x**2 + 5*x - 2)))/72

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Mathematica [A]  time = 0.0330053, size = 40, normalized size = 1.03 \[ \left (\frac{x}{2}-\frac{5}{12}\right ) \sqrt{-3 x^2+5 x-2}-\frac{\sin ^{-1}(5-6 x)}{24 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[-2 + 5*x - 3*x^2],x]

[Out]

(-5/12 + x/2)*Sqrt[-2 + 5*x - 3*x^2] - ArcSin[5 - 6*x]/(24*Sqrt[3])

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Maple [A]  time = 0.005, size = 32, normalized size = 0.8 \[{\frac{\arcsin \left ( -5+6\,x \right ) \sqrt{3}}{72}}-{\frac{5-6\,x}{12}\sqrt{-3\,{x}^{2}+5\,x-2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-3*x^2+5*x-2)^(1/2),x)

[Out]

1/72*arcsin(-5+6*x)*3^(1/2)-1/12*(5-6*x)*(-3*x^2+5*x-2)^(1/2)

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Maxima [A]  time = 0.864663, size = 55, normalized size = 1.41 \[ \frac{1}{2} \, \sqrt{-3 \, x^{2} + 5 \, x - 2} x + \frac{1}{72} \, \sqrt{3} \arcsin \left (6 \, x - 5\right ) - \frac{5}{12} \, \sqrt{-3 \, x^{2} + 5 \, x - 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-3*x^2 + 5*x - 2),x, algorithm="maxima")

[Out]

1/2*sqrt(-3*x^2 + 5*x - 2)*x + 1/72*sqrt(3)*arcsin(6*x - 5) - 5/12*sqrt(-3*x^2 +
5*x - 2)

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Fricas [A]  time = 0.223178, size = 69, normalized size = 1.77 \[ \frac{1}{72} \, \sqrt{3}{\left (2 \, \sqrt{3} \sqrt{-3 \, x^{2} + 5 \, x - 2}{\left (6 \, x - 5\right )} + \arctan \left (\frac{\sqrt{3}{\left (6 \, x - 5\right )}}{6 \, \sqrt{-3 \, x^{2} + 5 \, x - 2}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-3*x^2 + 5*x - 2),x, algorithm="fricas")

[Out]

1/72*sqrt(3)*(2*sqrt(3)*sqrt(-3*x^2 + 5*x - 2)*(6*x - 5) + arctan(1/6*sqrt(3)*(6
*x - 5)/sqrt(-3*x^2 + 5*x - 2)))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{- 3 x^{2} + 5 x - 2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-3*x**2+5*x-2)**(1/2),x)

[Out]

Integral(sqrt(-3*x**2 + 5*x - 2), x)

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GIAC/XCAS [A]  time = 0.209828, size = 42, normalized size = 1.08 \[ \frac{1}{12} \, \sqrt{-3 \, x^{2} + 5 \, x - 2}{\left (6 \, x - 5\right )} + \frac{1}{72} \, \sqrt{3} \arcsin \left (6 \, x - 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-3*x^2 + 5*x - 2),x, algorithm="giac")

[Out]

1/12*sqrt(-3*x^2 + 5*x - 2)*(6*x - 5) + 1/72*sqrt(3)*arcsin(6*x - 5)