3.567 \(\int \sqrt{1-\sqrt{x}-x} \, dx\)

Optimal. Leaf size=70 \[ -\frac{2}{3} \left (-x-\sqrt{x}+1\right )^{3/2}-\frac{1}{4} \left (2 \sqrt{x}+1\right ) \sqrt{-x-\sqrt{x}+1}-\frac{5}{8} \sin ^{-1}\left (\frac{2 \sqrt{x}+1}{\sqrt{5}}\right ) \]

[Out]

-((1 + 2*Sqrt[x])*Sqrt[1 - Sqrt[x] - x])/4 - (2*(1 - Sqrt[x] - x)^(3/2))/3 - (5*
ArcSin[(1 + 2*Sqrt[x])/Sqrt[5]])/8

_______________________________________________________________________________________

Rubi [A]  time = 0.0702571, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312 \[ -\frac{2}{3} \left (-x-\sqrt{x}+1\right )^{3/2}-\frac{1}{4} \left (2 \sqrt{x}+1\right ) \sqrt{-x-\sqrt{x}+1}-\frac{5}{8} \sin ^{-1}\left (\frac{2 \sqrt{x}+1}{\sqrt{5}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 - Sqrt[x] - x],x]

[Out]

-((1 + 2*Sqrt[x])*Sqrt[1 - Sqrt[x] - x])/4 - (2*(1 - Sqrt[x] - x)^(3/2))/3 - (5*
ArcSin[(1 + 2*Sqrt[x])/Sqrt[5]])/8

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.67464, size = 68, normalized size = 0.97 \[ - \frac{\left (2 \sqrt{x} + 1\right ) \sqrt{- \sqrt{x} - x + 1}}{4} - \frac{2 \left (- \sqrt{x} - x + 1\right )^{\frac{3}{2}}}{3} - \frac{5 \operatorname{atan}{\left (- \frac{- 2 \sqrt{x} - 1}{2 \sqrt{- \sqrt{x} - x + 1}} \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*sqrt(x) + 1)*sqrt(-sqrt(x) - x + 1)/4 - 2*(-sqrt(x) - x + 1)**(3/2)/3 - 5*at
an(-(-2*sqrt(x) - 1)/(2*sqrt(-sqrt(x) - x + 1)))/8

_______________________________________________________________________________________

Mathematica [A]  time = 0.0414724, size = 53, normalized size = 0.76 \[ \frac{1}{12} \sqrt{-x-\sqrt{x}+1} \left (8 x+2 \sqrt{x}-11\right )+\frac{5}{8} \sin ^{-1}\left (\frac{-2 \sqrt{x}-1}{\sqrt{5}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[1 - Sqrt[x] - x],x]

[Out]

(Sqrt[1 - Sqrt[x] - x]*(-11 + 2*Sqrt[x] + 8*x))/12 + (5*ArcSin[(-1 - 2*Sqrt[x])/
Sqrt[5]])/8

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 50, normalized size = 0.7 \[ -{\frac{2}{3} \left ( 1-x-\sqrt{x} \right ) ^{{\frac{3}{2}}}}+{\frac{1}{4} \left ( -2\,\sqrt{x}-1 \right ) \sqrt{1-x-\sqrt{x}}}-{\frac{5}{8}\arcsin \left ({\frac{2\,\sqrt{5}}{5} \left ( \sqrt{x}+{\frac{1}{2}} \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-x-x^(1/2))^(1/2),x)

[Out]

-2/3*(1-x-x^(1/2))^(3/2)+1/4*(-2*x^(1/2)-1)*(1-x-x^(1/2))^(1/2)-5/8*arcsin(2/5*5
^(1/2)*(x^(1/2)+1/2))

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{-x - \sqrt{x} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x - sqrt(x) + 1),x, algorithm="maxima")

[Out]

integrate(sqrt(-x - sqrt(x) + 1), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.977254, size = 82, normalized size = 1.17 \[ \frac{1}{12} \,{\left (8 \, x + 2 \, \sqrt{x} - 11\right )} \sqrt{-x - \sqrt{x} + 1} - \frac{5}{16} \, \arctan \left (\frac{8 \, x + 8 \, \sqrt{x} - 3}{4 \, \sqrt{-x - \sqrt{x} + 1}{\left (2 \, \sqrt{x} + 1\right )}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x - sqrt(x) + 1),x, algorithm="fricas")

[Out]

1/12*(8*x + 2*sqrt(x) - 11)*sqrt(-x - sqrt(x) + 1) - 5/16*arctan(1/4*(8*x + 8*sq
rt(x) - 3)/(sqrt(-x - sqrt(x) + 1)*(2*sqrt(x) + 1)))

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{- \sqrt{x} - x + 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(-sqrt(x) - x + 1), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.274965, size = 59, normalized size = 0.84 \[ \frac{1}{12} \,{\left (2 \, \sqrt{x}{\left (4 \, \sqrt{x} + 1\right )} - 11\right )} \sqrt{-x - \sqrt{x} + 1} - \frac{5}{8} \, \arcsin \left (\frac{1}{5} \, \sqrt{5}{\left (2 \, \sqrt{x} + 1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x - sqrt(x) + 1),x, algorithm="giac")

[Out]

1/12*(2*sqrt(x)*(4*sqrt(x) + 1) - 11)*sqrt(-x - sqrt(x) + 1) - 5/8*arcsin(1/5*sq
rt(5)*(2*sqrt(x) + 1))