Integrand size = 22, antiderivative size = 95 \[ \int \sqrt {1-\sqrt {x}-x} \sqrt {x} \, dx=\frac {9}{32} \left (1+2 \sqrt {x}\right ) \sqrt {1-\sqrt {x}-x}+\frac {5}{12} \left (1-\sqrt {x}-x\right )^{3/2}-\frac {1}{2} \left (1-\sqrt {x}-x\right )^{3/2} \sqrt {x}+\frac {45}{64} \arcsin \left (\frac {1+2 \sqrt {x}}{\sqrt {5}}\right ) \] Output:
9/32*(1+2*x^(1/2))*(1-x^(1/2)-x)^(1/2)+5/12*(1-x^(1/2)-x)^(3/2)-1/2*(1-x^( 1/2)-x)^(3/2)*x^(1/2)+45/64*arcsin(1/5*(1+2*x^(1/2))*5^(1/2))
Time = 0.20 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.75 \[ \int \sqrt {1-\sqrt {x}-x} \sqrt {x} \, dx=\frac {1}{96} \sqrt {1-\sqrt {x}-x} \left (67-34 \sqrt {x}+8 x+48 x^{3/2}\right )+\frac {45}{32} \arctan \left (\frac {\sqrt {x}}{-1+\sqrt {1-\sqrt {x}-x}}\right ) \] Input:
Integrate[Sqrt[1 - Sqrt[x] - x]*Sqrt[x],x]
Output:
(Sqrt[1 - Sqrt[x] - x]*(67 - 34*Sqrt[x] + 8*x + 48*x^(3/2)))/96 + (45*ArcT an[Sqrt[x]/(-1 + Sqrt[1 - Sqrt[x] - x])])/32
Time = 0.41 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.13, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {1693, 1166, 27, 1160, 1087, 1090, 223}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sqrt {-x-\sqrt {x}+1} \sqrt {x} \, dx\) |
\(\Big \downarrow \) 1693 |
\(\displaystyle 2 \int \sqrt {-x-\sqrt {x}+1} xd\sqrt {x}\) |
\(\Big \downarrow \) 1166 |
\(\displaystyle 2 \left (-\frac {1}{4} \int -\frac {1}{2} \left (2-5 \sqrt {x}\right ) \sqrt {-x-\sqrt {x}+1}d\sqrt {x}-\frac {1}{4} \sqrt {x} \left (-x-\sqrt {x}+1\right )^{3/2}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \left (\frac {1}{8} \int \left (2-5 \sqrt {x}\right ) \sqrt {-x-\sqrt {x}+1}d\sqrt {x}-\frac {1}{4} \left (-x-\sqrt {x}+1\right )^{3/2} \sqrt {x}\right )\) |
\(\Big \downarrow \) 1160 |
\(\displaystyle 2 \left (\frac {1}{8} \left (\frac {9}{2} \int \sqrt {-x-\sqrt {x}+1}d\sqrt {x}+\frac {5}{3} \left (-x-\sqrt {x}+1\right )^{3/2}\right )-\frac {1}{4} \left (-x-\sqrt {x}+1\right )^{3/2} \sqrt {x}\right )\) |
\(\Big \downarrow \) 1087 |
\(\displaystyle 2 \left (\frac {1}{8} \left (\frac {9}{2} \left (\frac {5}{8} \int \frac {1}{\sqrt {-x-\sqrt {x}+1}}d\sqrt {x}+\frac {1}{4} \sqrt {-x-\sqrt {x}+1} \left (2 \sqrt {x}+1\right )\right )+\frac {5}{3} \left (-x-\sqrt {x}+1\right )^{3/2}\right )-\frac {1}{4} \left (-x-\sqrt {x}+1\right )^{3/2} \sqrt {x}\right )\) |
\(\Big \downarrow \) 1090 |
\(\displaystyle 2 \left (\frac {1}{8} \left (\frac {9}{2} \left (\frac {1}{4} \left (2 \sqrt {x}+1\right ) \sqrt {-x-\sqrt {x}+1}-\frac {1}{8} \sqrt {5} \int \frac {1}{\sqrt {1-\frac {x}{5}}}d\left (-2 \sqrt {x}-1\right )\right )+\frac {5}{3} \left (-x-\sqrt {x}+1\right )^{3/2}\right )-\frac {1}{4} \left (-x-\sqrt {x}+1\right )^{3/2} \sqrt {x}\right )\) |
\(\Big \downarrow \) 223 |
\(\displaystyle 2 \left (\frac {1}{8} \left (\frac {9}{2} \left (\frac {1}{4} \left (2 \sqrt {x}+1\right ) \sqrt {-x-\sqrt {x}+1}-\frac {5}{8} \arcsin \left (\frac {-2 \sqrt {x}-1}{\sqrt {5}}\right )\right )+\frac {5}{3} \left (-x-\sqrt {x}+1\right )^{3/2}\right )-\frac {1}{4} \left (-x-\sqrt {x}+1\right )^{3/2} \sqrt {x}\right )\) |
Input:
Int[Sqrt[1 - Sqrt[x] - x]*Sqrt[x],x]
Output:
2*(-1/4*((1 - Sqrt[x] - x)^(3/2)*Sqrt[x]) + ((5*(1 - Sqrt[x] - x)^(3/2))/3 + (9*(((1 + 2*Sqrt[x])*Sqrt[1 - Sqrt[x] - x])/4 - (5*ArcSin[(-1 - 2*Sqrt[ x])/Sqrt[5]])/8))/2)/8)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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] - Simp[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] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/(2*c*(-4* (c/(b^2 - 4*a*c)))^p) Subst[Int[Simp[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]
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] + Simp[(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[p, -1]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Simp[1/(c*(m + 2*p + 1)) Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]* (a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && If[Ration alQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadrat icQ[a, b, c, d, e, m, p, x]
Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol ] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && IntegerQ [Simplify[(m + 1)/n]]
Time = 0.02 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.71
method | result | size |
derivativedivides | \(-\frac {\left (1-\sqrt {x}-x \right )^{\frac {3}{2}} \sqrt {x}}{2}+\frac {5 \left (1-\sqrt {x}-x \right )^{\frac {3}{2}}}{12}-\frac {9 \left (-2 \sqrt {x}-1\right ) \sqrt {1-\sqrt {x}-x}}{32}+\frac {45 \arcsin \left (\frac {2 \sqrt {5}\, \left (\sqrt {x}+\frac {1}{2}\right )}{5}\right )}{64}\) | \(67\) |
default | \(-\frac {\left (1-\sqrt {x}-x \right )^{\frac {3}{2}} \sqrt {x}}{2}+\frac {5 \left (1-\sqrt {x}-x \right )^{\frac {3}{2}}}{12}-\frac {9 \left (-2 \sqrt {x}-1\right ) \sqrt {1-\sqrt {x}-x}}{32}+\frac {45 \arcsin \left (\frac {2 \sqrt {5}\, \left (\sqrt {x}+\frac {1}{2}\right )}{5}\right )}{64}\) | \(67\) |
Input:
int((1-x^(1/2)-x)^(1/2)*x^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/2*(1-x^(1/2)-x)^(3/2)*x^(1/2)+5/12*(1-x^(1/2)-x)^(3/2)-9/32*(-2*x^(1/2) -1)*(1-x^(1/2)-x)^(1/2)+45/64*arcsin(2/5*5^(1/2)*(x^(1/2)+1/2))
Time = 0.78 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.78 \[ \int \sqrt {1-\sqrt {x}-x} \sqrt {x} \, dx=\frac {1}{96} \, {\left (2 \, {\left (24 \, x - 17\right )} \sqrt {x} + 8 \, x + 67\right )} \sqrt {-x - \sqrt {x} + 1} - \frac {45}{128} \, \arctan \left (\frac {4 \, {\left (2 \, {\left (8 \, x - 7\right )} \sqrt {x} - 8 \, x - 3\right )} \sqrt {-x - \sqrt {x} + 1}}{64 \, x^{2} - 112 \, x + 9}\right ) \] Input:
integrate((1-x^(1/2)-x)^(1/2)*x^(1/2),x, algorithm="fricas")
Output:
1/96*(2*(24*x - 17)*sqrt(x) + 8*x + 67)*sqrt(-x - sqrt(x) + 1) - 45/128*ar ctan(4*(2*(8*x - 7)*sqrt(x) - 8*x - 3)*sqrt(-x - sqrt(x) + 1)/(64*x^2 - 11 2*x + 9))
Time = 0.39 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.59 \[ \int \sqrt {1-\sqrt {x}-x} \sqrt {x} \, dx=2 \sqrt {- \sqrt {x} - x + 1} \left (\frac {x^{\frac {3}{2}}}{4} - \frac {17 \sqrt {x}}{96} + \frac {x}{24} + \frac {67}{192}\right ) + \frac {45 \operatorname {asin}{\left (\frac {2 \sqrt {5} \left (\sqrt {x} + \frac {1}{2}\right )}{5} \right )}}{64} \] Input:
integrate((1-x**(1/2)-x)**(1/2)*x**(1/2),x)
Output:
2*sqrt(-sqrt(x) - x + 1)*(x**(3/2)/4 - 17*sqrt(x)/96 + x/24 + 67/192) + 45 *asin(2*sqrt(5)*(sqrt(x) + 1/2)/5)/64
\[ \int \sqrt {1-\sqrt {x}-x} \sqrt {x} \, dx=\int { \sqrt {x} \sqrt {-x - \sqrt {x} + 1} \,d x } \] Input:
integrate((1-x^(1/2)-x)^(1/2)*x^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(x)*sqrt(-x - sqrt(x) + 1), x)
Time = 0.14 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.54 \[ \int \sqrt {1-\sqrt {x}-x} \sqrt {x} \, dx=\frac {1}{96} \, {\left (2 \, {\left (4 \, \sqrt {x} {\left (6 \, \sqrt {x} + 1\right )} - 17\right )} \sqrt {x} + 67\right )} \sqrt {-x - \sqrt {x} + 1} + \frac {45}{64} \, \arcsin \left (\frac {1}{5} \, \sqrt {5} {\left (2 \, \sqrt {x} + 1\right )}\right ) \] Input:
integrate((1-x^(1/2)-x)^(1/2)*x^(1/2),x, algorithm="giac")
Output:
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))
Timed out. \[ \int \sqrt {1-\sqrt {x}-x} \sqrt {x} \, dx=\int \sqrt {x}\,\sqrt {1-\sqrt {x}-x} \,d x \] Input:
int(x^(1/2)*(1 - x^(1/2) - x)^(1/2),x)
Output:
int(x^(1/2)*(1 - x^(1/2) - x)^(1/2), x)
Time = 0.18 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.77 \[ \int \sqrt {1-\sqrt {x}-x} \sqrt {x} \, dx=\frac {45 \mathit {asin} \left (\frac {2 \sqrt {x}+1}{\sqrt {5}}\right )}{64}+\frac {\sqrt {x}\, \sqrt {-\sqrt {x}-x +1}\, x}{2}-\frac {17 \sqrt {x}\, \sqrt {-\sqrt {x}-x +1}}{48}+\frac {\sqrt {-\sqrt {x}-x +1}\, x}{12}+\frac {67 \sqrt {-\sqrt {x}-x +1}}{96}-\frac {5 \sqrt {5}}{12} \] Input:
int((1-x^(1/2)-x)^(1/2)*x^(1/2),x)
Output:
(135*asin((2*sqrt(x) + 1)/sqrt(5)) + 96*sqrt(x)*sqrt( - sqrt(x) - x + 1)*x - 68*sqrt(x)*sqrt( - sqrt(x) - x + 1) + 16*sqrt( - sqrt(x) - x + 1)*x + 1 34*sqrt( - sqrt(x) - x + 1) - 80*sqrt(5))/192