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

Optimal. Leaf size=95 \[ -\frac{1}{2} \sqrt{x} \left (-x-\sqrt{x}+1\right )^{3/2}+\frac{5}{12} \left (-x-\sqrt{x}+1\right )^{3/2}+\frac{9}{32} \left (2 \sqrt{x}+1\right ) \sqrt{-x-\sqrt{x}+1}+\frac{45}{64} \sin ^{-1}\left (\frac{2 \sqrt{x}+1}{\sqrt{5}}\right ) \]

[Out]

(9*(1 + 2*Sqrt[x])*Sqrt[1 - Sqrt[x] - x])/32 + (5*(1 - Sqrt[x] - x)^(3/2))/12 -
((1 - Sqrt[x] - x)^(3/2)*Sqrt[x])/2 + (45*ArcSin[(1 + 2*Sqrt[x])/Sqrt[5]])/64

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Rubi [A]  time = 0.111151, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{1}{2} \sqrt{x} \left (-x-\sqrt{x}+1\right )^{3/2}+\frac{5}{12} \left (-x-\sqrt{x}+1\right )^{3/2}+\frac{9}{32} \left (2 \sqrt{x}+1\right ) \sqrt{-x-\sqrt{x}+1}+\frac{45}{64} \sin ^{-1}\left (\frac{2 \sqrt{x}+1}{\sqrt{5}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 - Sqrt[x] - x]*Sqrt[x],x]

[Out]

(9*(1 + 2*Sqrt[x])*Sqrt[1 - Sqrt[x] - x])/32 + (5*(1 - Sqrt[x] - x)^(3/2))/12 -
((1 - Sqrt[x] - x)^(3/2)*Sqrt[x])/2 + (45*ArcSin[(1 + 2*Sqrt[x])/Sqrt[5]])/64

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Rubi in Sympy [A]  time = 6.77057, size = 87, normalized size = 0.92 \[ - \frac{\sqrt{x} \left (- \sqrt{x} - x + 1\right )^{\frac{3}{2}}}{2} + \frac{9 \left (2 \sqrt{x} + 1\right ) \sqrt{- \sqrt{x} - x + 1}}{32} + \frac{5 \left (- \sqrt{x} - x + 1\right )^{\frac{3}{2}}}{12} + \frac{45 \operatorname{atan}{\left (- \frac{- 2 \sqrt{x} - 1}{2 \sqrt{- \sqrt{x} - x + 1}} \right )}}{64} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(x)*(-sqrt(x) - x + 1)**(3/2)/2 + 9*(2*sqrt(x) + 1)*sqrt(-sqrt(x) - x + 1)/
32 + 5*(-sqrt(x) - x + 1)**(3/2)/12 + 45*atan(-(-2*sqrt(x) - 1)/(2*sqrt(-sqrt(x)
- x + 1)))/64

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Mathematica [A]  time = 0.0483693, size = 60, normalized size = 0.63 \[ \frac{1}{96} \sqrt{-x-\sqrt{x}+1} \left (48 x^{3/2}+8 x-34 \sqrt{x}+67\right )-\frac{45}{64} \sin ^{-1}\left (\frac{-2 \sqrt{x}-1}{\sqrt{5}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - Sqrt[x] - x]*(67 - 34*Sqrt[x] + 8*x + 48*x^(3/2)))/96 - (45*ArcSin[(-1
- 2*Sqrt[x])/Sqrt[5]])/64

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Maple [A]  time = 0.005, size = 67, normalized size = 0.7 \[ -{\frac{1}{2} \left ( 1-x-\sqrt{x} \right ) ^{{\frac{3}{2}}}\sqrt{x}}+{\frac{5}{12} \left ( 1-x-\sqrt{x} \right ) ^{{\frac{3}{2}}}}-{\frac{9}{32} \left ( -2\,\sqrt{x}-1 \right ) \sqrt{1-x-\sqrt{x}}}+{\frac{45}{64}\arcsin \left ({\frac{2\,\sqrt{5}}{5} \left ( \sqrt{x}+{\frac{1}{2}} \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(1-x-x^(1/2))^(3/2)*x^(1/2)+5/12*(1-x-x^(1/2))^(3/2)-9/32*(-2*x^(1/2)-1)*(1
-x-x^(1/2))^(1/2)+45/64*arcsin(2/5*5^(1/2)*(x^(1/2)+1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(x)*sqrt(-x - sqrt(x) + 1), x)

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Fricas [A]  time = 1.22884, size = 89, normalized size = 0.94 \[ \frac{1}{96} \,{\left (2 \,{\left (24 \, x - 17\right )} \sqrt{x} + 8 \, x + 67\right )} \sqrt{-x - \sqrt{x} + 1} + \frac{45}{128} \, \arctan \left (\frac{8 \, x + 8 \, \sqrt{x} - 3}{4 \, \sqrt{-x - \sqrt{x} + 1}{\left (2 \, \sqrt{x} + 1\right )}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/96*(2*(24*x - 17)*sqrt(x) + 8*x + 67)*sqrt(-x - sqrt(x) + 1) + 45/128*arctan(1
/4*(8*x + 8*sqrt(x) - 3)/(sqrt(-x - sqrt(x) + 1)*(2*sqrt(x) + 1)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(x)*sqrt(-sqrt(x) - x + 1), x)

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GIAC/XCAS [A]  time = 0.268708, size = 69, normalized size = 0.73 \[ \frac{1}{96} \,{\left (2 \,{\left (4 \, \sqrt{x}{\left (6 \, \sqrt{x} + 1\right )} - 17\right )} \sqrt{x} + 67\right )} \sqrt{-x - \sqrt{x} + 1} + \frac{45}{64} \, \arcsin \left (\frac{1}{5} \, \sqrt{5}{\left (2 \, \sqrt{x} + 1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/96*(2*(4*sqrt(x)*(6*sqrt(x) + 1) - 17)*sqrt(x) + 67)*sqrt(-x - sqrt(x) + 1) +
45/64*arcsin(1/5*sqrt(5)*(2*sqrt(x) + 1))