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

Optimal. Leaf size=26 \[ -\sqrt{4 x-x^2}-4 \sin ^{-1}\left (1-\frac{x}{2}\right ) \]

[Out]

-Sqrt[4*x - x^2] - 4*ArcSin[1 - x/2]

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Rubi [A]  time = 0.0312822, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ -\sqrt{4 x-x^2}-4 \sin ^{-1}\left (1-\frac{x}{2}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(2 + x)/Sqrt[4*x - x^2],x]

[Out]

-Sqrt[4*x - x^2] - 4*ArcSin[1 - x/2]

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Rubi in Sympy [A]  time = 2.16481, size = 17, normalized size = 0.65 \[ - \sqrt{- x^{2} + 4 x} + 4 \operatorname{asin}{\left (\frac{x}{2} - 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-x**2 + 4*x) + 4*asin(x/2 - 1)

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Mathematica [A]  time = 0.0284919, size = 45, normalized size = 1.73 \[ \frac{(x-4) x+8 \sqrt{x-4} \sqrt{x} \log \left (\sqrt{x-4}+\sqrt{x}\right )}{\sqrt{-(x-4) x}} \]

Antiderivative was successfully verified.

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

[Out]

((-4 + x)*x + 8*Sqrt[-4 + x]*Sqrt[x]*Log[Sqrt[-4 + x] + Sqrt[x]])/Sqrt[-((-4 + x
)*x)]

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Maple [A]  time = 0.009, size = 23, normalized size = 0.9 \[ 4\,\arcsin \left ( x/2-1 \right ) -\sqrt{-{x}^{2}+4\,x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+x)/(-x^2+4*x)^(1/2),x)

[Out]

4*arcsin(1/2*x-1)-(-x^2+4*x)^(1/2)

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Maxima [A]  time = 0.786281, size = 30, normalized size = 1.15 \[ -\sqrt{-x^{2} + 4 \, x} - 4 \, \arcsin \left (-\frac{1}{2} \, x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.264813, size = 43, normalized size = 1.65 \[ -\sqrt{-x^{2} + 4 \, x} - 8 \, \arctan \left (\frac{\sqrt{-x^{2} + 4 \, x}}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-x^2 + 4*x) - 8*arctan(sqrt(-x^2 + 4*x)/x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((x + 2)/sqrt(-x*(x - 4)), x)

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GIAC/XCAS [A]  time = 0.266707, size = 30, normalized size = 1.15 \[ -\sqrt{-x^{2} + 4 \, x} + 4 \, \arcsin \left (\frac{1}{2} \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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