3.2389 \(\int \frac{3+x}{\sqrt{5-4 x-x^2}} \, dx\)

Optimal. Leaf size=29 \[ -\sqrt{-x^2-4 x+5}-\sin ^{-1}\left (\frac{1}{3} (-x-2)\right ) \]

[Out]

-Sqrt[5 - 4*x - x^2] - ArcSin[(-2 - x)/3]

_______________________________________________________________________________________

Rubi [A]  time = 0.0374607, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\sqrt{-x^2-4 x+5}-\sin ^{-1}\left (\frac{1}{3} (-x-2)\right ) \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[5 - 4*x - x^2] - ArcSin[(-2 - x)/3]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.3088, size = 34, normalized size = 1.17 \[ - \sqrt{- x^{2} - 4 x + 5} + \operatorname{atan}{\left (- \frac{- 2 x - 4}{2 \sqrt{- x^{2} - 4 x + 5}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-x**2 - 4*x + 5) + atan(-(-2*x - 4)/(2*sqrt(-x**2 - 4*x + 5)))

_______________________________________________________________________________________

Mathematica [A]  time = 0.02356, size = 25, normalized size = 0.86 \[ \sin ^{-1}\left (\frac{x+2}{3}\right )-\sqrt{-x^2-4 x+5} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[5 - 4*x - x^2] + ArcSin[(2 + x)/3]

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 22, normalized size = 0.8 \[ \arcsin \left ({\frac{2}{3}}+{\frac{x}{3}} \right ) -\sqrt{-{x}^{2}-4\,x+5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3+x)/(-x^2-4*x+5)^(1/2),x)

[Out]

arcsin(2/3+1/3*x)-(-x^2-4*x+5)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.749664, size = 31, normalized size = 1.07 \[ -\sqrt{-x^{2} - 4 \, x + 5} - \arcsin \left (-\frac{1}{3} \, x - \frac{2}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-x^2 - 4*x + 5) - arcsin(-1/3*x - 2/3)

_______________________________________________________________________________________

Fricas [A]  time = 0.22468, size = 43, normalized size = 1.48 \[ -\sqrt{-x^{2} - 4 \, x + 5} + \arctan \left (\frac{x + 2}{\sqrt{-x^{2} - 4 \, x + 5}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-x^2 - 4*x + 5) + arctan((x + 2)/sqrt(-x^2 - 4*x + 5))

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x + 3}{\sqrt{- \left (x - 1\right ) \left (x + 5\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((x + 3)/sqrt(-(x - 1)*(x + 5)), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.210232, size = 28, normalized size = 0.97 \[ -\sqrt{-x^{2} - 4 \, x + 5} + \arcsin \left (\frac{1}{3} \, x + \frac{2}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-x^2 - 4*x + 5) + arcsin(1/3*x + 2/3)