3.2390 \(\int \frac{5-4 x}{\sqrt{-8+12 x-4 x^2}} \, dx\)

Optimal. Leaf size=25 \[ \sqrt{-4 x^2+12 x-8}+\frac{1}{2} \sin ^{-1}(3-2 x) \]

[Out]

Sqrt[-8 + 12*x - 4*x^2] + ArcSin[3 - 2*x]/2

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Rubi [A]  time = 0.0342945, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \sqrt{-4 x^2+12 x-8}+\frac{1}{2} \sin ^{-1}(3-2 x) \]

Antiderivative was successfully verified.

[In]  Int[(5 - 4*x)/Sqrt[-8 + 12*x - 4*x^2],x]

[Out]

Sqrt[-8 + 12*x - 4*x^2] + ArcSin[3 - 2*x]/2

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Rubi in Sympy [A]  time = 6.41453, size = 34, normalized size = 1.36 \[ 2 \sqrt{- x^{2} + 3 x - 2} + \frac{\operatorname{atan}{\left (\frac{- 2 x + 3}{2 \sqrt{- x^{2} + 3 x - 2}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/2*(5-4*x)/(-x**2+3*x-2)**(1/2),x)

[Out]

2*sqrt(-x**2 + 3*x - 2) + atan((-2*x + 3)/(2*sqrt(-x**2 + 3*x - 2)))/2

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Mathematica [A]  time = 0.0316409, size = 27, normalized size = 1.08 \[ 2 \sqrt{-x^2+3 x-2}+\frac{1}{2} \sin ^{-1}(3-2 x) \]

Antiderivative was successfully verified.

[In]  Integrate[(5 - 4*x)/Sqrt[-8 + 12*x - 4*x^2],x]

[Out]

2*Sqrt[-2 + 3*x - x^2] + ArcSin[3 - 2*x]/2

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Maple [A]  time = 0.013, size = 24, normalized size = 1. \[ -{\frac{\arcsin \left ( -3+2\,x \right ) }{2}}+2\,\sqrt{-{x}^{2}+3\,x-2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/2*(5-4*x)/(-x^2+3*x-2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.751059, size = 31, normalized size = 1.24 \[ 2 \, \sqrt{-x^{2} + 3 \, x - 2} - \frac{1}{2} \, \arcsin \left (2 \, x - 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/2*(4*x - 5)/sqrt(-x^2 + 3*x - 2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.23748, size = 50, normalized size = 2. \[ 2 \, \sqrt{-x^{2} + 3 \, x - 2} - \frac{1}{2} \, \arctan \left (\frac{2 \, x - 3}{2 \, \sqrt{-x^{2} + 3 \, x - 2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/2*(4*x - 5)/sqrt(-x^2 + 3*x - 2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{\int \frac{4 x}{\sqrt{- x^{2} + 3 x - 2}}\, dx + \int \left (- \frac{5}{\sqrt{- x^{2} + 3 x - 2}}\right )\, dx}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/2*(5-4*x)/(-x**2+3*x-2)**(1/2),x)

[Out]

-(Integral(4*x/sqrt(-x**2 + 3*x - 2), x) + Integral(-5/sqrt(-x**2 + 3*x - 2), x)
)/2

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GIAC/XCAS [A]  time = 0.210784, size = 31, normalized size = 1.24 \[ 2 \, \sqrt{-x^{2} + 3 \, x - 2} - \frac{1}{2} \, \arcsin \left (2 \, x - 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/2*(4*x - 5)/sqrt(-x^2 + 3*x - 2),x, algorithm="giac")

[Out]

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