3.8.23 \(\int \sqrt {1-\sqrt {x}-x} \, dx\) [723]

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

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {1355, 654, 626, 633, 222} \begin {gather*} -\frac {5}{8} \text {ArcSin}\left (\frac {2 \sqrt {x}+1}{\sqrt {5}}\right )-\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} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - Sqrt[x] - x],x]

[Out]

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

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 626

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x)*((a + b*x + c*x^2)^p/(2*c*(2*p + 1
))), x] - Dist[p*((b^2 - 4*a*c)/(2*c*(2*p + 1))), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 633

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*(-4*(c/(b^2 - 4*a*c)))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 654

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*((a + b*x + c*x^2)^(p +
1)/(2*c*(p + 1))), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 1355

Int[((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Dist[k, Subst[I
nt[x^(k - 1)*(a + b*x^(k*n) + c*x^(2*k*n))^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && EqQ[n2, 2*n] &
& FractionQ[n]

Rubi steps

\begin {align*} \int \sqrt {1-\sqrt {x}-x} \, dx &=2 \text {Subst}\left (\int x \sqrt {1-x-x^2} \, dx,x,\sqrt {x}\right )\\ &=-\frac {2}{3} \left (1-\sqrt {x}-x\right )^{3/2}-\text {Subst}\left (\int \sqrt {1-x-x^2} \, dx,x,\sqrt {x}\right )\\ &=-\frac {1}{4} \left (1+2 \sqrt {x}\right ) \sqrt {1-\sqrt {x}-x}-\frac {2}{3} \left (1-\sqrt {x}-x\right )^{3/2}-\frac {5}{8} \text {Subst}\left (\int \frac {1}{\sqrt {1-x-x^2}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {1}{4} \left (1+2 \sqrt {x}\right ) \sqrt {1-\sqrt {x}-x}-\frac {2}{3} \left (1-\sqrt {x}-x\right )^{3/2}+\frac {1}{8} \sqrt {5} \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{5}}} \, dx,x,-1-2 \sqrt {x}\right )\\ &=-\frac {1}{4} \left (1+2 \sqrt {x}\right ) \sqrt {1-\sqrt {x}-x}-\frac {2}{3} \left (1-\sqrt {x}-x\right )^{3/2}-\frac {5}{8} \sin ^{-1}\left (\frac {1+2 \sqrt {x}}{\sqrt {5}}\right )\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 64, normalized size = 0.91 \begin {gather*} \frac {1}{12} \sqrt {1-\sqrt {x}-x} \left (-11+2 \sqrt {x}+8 x\right )-\frac {5}{4} \tan ^{-1}\left (\frac {\sqrt {x}}{-1+\sqrt {1-\sqrt {x}-x}}\right ) \end {gather*}

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*ArcTan[Sqrt[x]/(-1 + Sqrt[1 - Sqrt[x] - x])])/4

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Maple [A]
time = 0.02, size = 50, normalized size = 0.71

method	result	size
derivativedivides	\(-\frac {2 \left (1-x -\sqrt {x}\right )^{\frac {3}{2}}}{3}+\frac {\left (-2 \sqrt {x}-1\right ) \sqrt {1-x -\sqrt {x}}}{4}-\frac {5 \arcsin \left (\frac {2 \sqrt {5}\, \left (\sqrt {x}+\frac {1}{2}\right )}{5}\right )}{8}\)	\(50\)
default	\(-\frac {2 \left (1-x -\sqrt {x}\right )^{\frac {3}{2}}}{3}+\frac {\left (-2 \sqrt {x}-1\right ) \sqrt {1-x -\sqrt {x}}}{4}-\frac {5 \arcsin \left (\frac {2 \sqrt {5}\, \left (\sqrt {x}+\frac {1}{2}\right )}{5}\right )}{8}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-x-x^(1/2))^(1/2),x,method=_RETURNVERBOSE)

[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))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x-x^(1/2))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(-x - sqrt(x) + 1), x)

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Fricas [A]
time = 0.83, size = 84, normalized size = 1.20 \begin {gather*} \frac {1}{12} \, {\left (8 \, x + 2 \, \sqrt {x} - 11\right )} \sqrt {-x - \sqrt {x} + 1} + \frac {5}{16} \, \arctan \left (-\frac {{\left (8 \, x^{2} - {\left (16 \, x^{2} - 38 \, x + 11\right )} \sqrt {x} - 9 \, x + 3\right )} \sqrt {-x - \sqrt {x} + 1}}{4 \, {\left (4 \, x^{3} - 13 \, x^{2} + 7 \, x - 1\right )}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x-x^(1/2))^(1/2),x, algorithm="fricas")

[Out]

1/12*(8*x + 2*sqrt(x) - 11)*sqrt(-x - sqrt(x) + 1) + 5/16*arctan(-1/4*(8*x^2 - (16*x^2 - 38*x + 11)*sqrt(x) -
9*x + 3)*sqrt(-x - sqrt(x) + 1)/(4*x^3 - 13*x^2 + 7*x - 1))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {- \sqrt {x} - x + 1}\, dx \end {gather*}

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)

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Giac [A]
time = 2.07, size = 44, normalized size = 0.63 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x-x^(1/2))^(1/2),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \sqrt {1-\sqrt {x}-x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x^(1/2) - x)^(1/2),x)

[Out]

int((1 - x^(1/2) - x)^(1/2), x)

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