Integrand size = 13, antiderivative size = 68 \[ \int \sqrt {\sqrt {-1+x}+x} \, dx=-\frac {1}{4} \left (1+2 \sqrt {-1+x}\right ) \sqrt {\sqrt {-1+x}+x}+\frac {2}{3} \left (\sqrt {-1+x}+x\right )^{3/2}-\frac {3}{8} \text {arcsinh}\left (\frac {1+2 \sqrt {-1+x}}{\sqrt {3}}\right ) \] Output:
-1/4*(1+2*(-1+x)^(1/2))*((-1+x)^(1/2)+x)^(1/2)+2/3*((-1+x)^(1/2)+x)^(3/2)- 3/8*arcsinh(1/3*(1+2*(-1+x)^(1/2))*3^(1/2))
Time = 0.19 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.96 \[ \int \sqrt {\sqrt {-1+x}+x} \, dx=\frac {1}{12} \left (5+2 \sqrt {-1+x}+8 (-1+x)\right ) \sqrt {\sqrt {-1+x}+x}+\frac {3}{8} \log \left (-1-2 \sqrt {-1+x}+2 \sqrt {\sqrt {-1+x}+x}\right ) \] Input:
Integrate[Sqrt[Sqrt[-1 + x] + x],x]
Output:
((5 + 2*Sqrt[-1 + x] + 8*(-1 + x))*Sqrt[Sqrt[-1 + x] + x])/12 + (3*Log[-1 - 2*Sqrt[-1 + x] + 2*Sqrt[Sqrt[-1 + x] + x]])/8
Time = 0.40 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.10, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {7267, 1160, 1087, 1090, 222}
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.
\(\displaystyle \int \sqrt {x+\sqrt {x-1}} \, dx\) |
\(\Big \downarrow \) 7267 |
\(\displaystyle 2 \int \sqrt {x-1} \sqrt {x+\sqrt {x-1}}d\sqrt {x-1}\) |
\(\Big \downarrow \) 1160 |
\(\displaystyle 2 \left (\frac {1}{3} \left (x+\sqrt {x-1}\right )^{3/2}-\frac {1}{2} \int \sqrt {x+\sqrt {x-1}}d\sqrt {x-1}\right )\) |
\(\Big \downarrow \) 1087 |
\(\displaystyle 2 \left (\frac {1}{2} \left (-\frac {3}{8} \int \frac {1}{\sqrt {x+\sqrt {x-1}}}d\sqrt {x-1}-\frac {1}{4} \sqrt {x+\sqrt {x-1}} \left (2 \sqrt {x-1}+1\right )\right )+\frac {1}{3} \left (x+\sqrt {x-1}\right )^{3/2}\right )\) |
\(\Big \downarrow \) 1090 |
\(\displaystyle 2 \left (\frac {1}{2} \left (-\frac {1}{8} \sqrt {3} \int \frac {1}{\sqrt {\frac {x-1}{3}+1}}d\left (2 \sqrt {x-1}+1\right )-\frac {1}{4} \sqrt {x+\sqrt {x-1}} \left (2 \sqrt {x-1}+1\right )\right )+\frac {1}{3} \left (x+\sqrt {x-1}\right )^{3/2}\right )\) |
\(\Big \downarrow \) 222 |
\(\displaystyle 2 \left (\frac {1}{2} \left (-\frac {3}{8} \text {arcsinh}\left (\frac {2 \sqrt {x-1}+1}{\sqrt {3}}\right )-\frac {1}{4} \sqrt {x+\sqrt {x-1}} \left (2 \sqrt {x-1}+1\right )\right )+\frac {1}{3} \left (x+\sqrt {x-1}\right )^{3/2}\right )\) |
Input:
Int[Sqrt[Sqrt[-1 + x] + x],x]
Output:
2*((Sqrt[-1 + x] + x)^(3/2)/3 + (-1/4*((1 + 2*Sqrt[-1 + x])*Sqrt[Sqrt[-1 + x] + x]) - (3*ArcSinh[(1 + 2*Sqrt[-1 + x])/Sqrt[3]])/8)/2)
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[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[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si mp[lst[[2]]*lst[[4]] Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x ] /; !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
Time = 0.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.71
method | result | size |
derivativedivides | \(\frac {2 \left (\sqrt {-1+x}+x \right )^{\frac {3}{2}}}{3}-\frac {\left (1+2 \sqrt {-1+x}\right ) \sqrt {\sqrt {-1+x}+x}}{4}-\frac {3 \,\operatorname {arcsinh}\left (\frac {2 \sqrt {3}\, \left (\sqrt {-1+x}+\frac {1}{2}\right )}{3}\right )}{8}\) | \(48\) |
default | \(\frac {2 \left (\sqrt {-1+x}+x \right )^{\frac {3}{2}}}{3}-\frac {\left (1+2 \sqrt {-1+x}\right ) \sqrt {\sqrt {-1+x}+x}}{4}-\frac {3 \,\operatorname {arcsinh}\left (\frac {2 \sqrt {3}\, \left (\sqrt {-1+x}+\frac {1}{2}\right )}{3}\right )}{8}\) | \(48\) |
Input:
int(((-1+x)^(1/2)+x)^(1/2),x,method=_RETURNVERBOSE)
Output:
2/3*((-1+x)^(1/2)+x)^(3/2)-1/4*(1+2*(-1+x)^(1/2))*((-1+x)^(1/2)+x)^(1/2)-3 /8*arcsinh(2/3*3^(1/2)*((-1+x)^(1/2)+1/2))
Time = 0.44 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.87 \[ \int \sqrt {\sqrt {-1+x}+x} \, dx=\frac {1}{12} \, {\left (8 \, x + 2 \, \sqrt {x - 1} - 3\right )} \sqrt {x + \sqrt {x - 1}} + \frac {3}{16} \, \log \left (-4 \, \sqrt {x + \sqrt {x - 1}} {\left (2 \, \sqrt {x - 1} + 1\right )} + 8 \, x + 8 \, \sqrt {x - 1} - 3\right ) \] Input:
integrate(((x-1)^(1/2)+x)^(1/2),x, algorithm="fricas")
Output:
1/12*(8*x + 2*sqrt(x - 1) - 3)*sqrt(x + sqrt(x - 1)) + 3/16*log(-4*sqrt(x + sqrt(x - 1))*(2*sqrt(x - 1) + 1) + 8*x + 8*sqrt(x - 1) - 3)
Time = 0.33 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.75 \[ \int \sqrt {\sqrt {-1+x}+x} \, dx=2 \sqrt {x + \sqrt {x - 1}} \left (\frac {x}{3} + \frac {\sqrt {x - 1}}{12} - \frac {1}{8}\right ) - \frac {3 \operatorname {asinh}{\left (\frac {2 \sqrt {3} \left (\sqrt {x - 1} + \frac {1}{2}\right )}{3} \right )}}{8} \] Input:
integrate(((x-1)**(1/2)+x)**(1/2),x)
Output:
2*sqrt(x + sqrt(x - 1))*(x/3 + sqrt(x - 1)/12 - 1/8) - 3*asinh(2*sqrt(3)*( sqrt(x - 1) + 1/2)/3)/8
\[ \int \sqrt {\sqrt {-1+x}+x} \, dx=\int { \sqrt {x + \sqrt {x - 1}} \,d x } \] Input:
integrate(((x-1)^(1/2)+x)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(x + sqrt(x - 1)), x)
Time = 0.14 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.78 \[ \int \sqrt {\sqrt {-1+x}+x} \, dx=\frac {1}{12} \, {\left (2 \, \sqrt {x - 1} {\left (4 \, \sqrt {x - 1} + 1\right )} + 5\right )} \sqrt {x + \sqrt {x - 1}} + \frac {3}{8} \, \log \left (2 \, \sqrt {x + \sqrt {x - 1}} - 2 \, \sqrt {x - 1} - 1\right ) \] Input:
integrate(((x-1)^(1/2)+x)^(1/2),x, algorithm="giac")
Output:
1/12*(2*sqrt(x - 1)*(4*sqrt(x - 1) + 1) + 5)*sqrt(x + sqrt(x - 1)) + 3/8*l og(2*sqrt(x + sqrt(x - 1)) - 2*sqrt(x - 1) - 1)
Timed out. \[ \int \sqrt {\sqrt {-1+x}+x} \, dx=\int \sqrt {x+\sqrt {x-1}} \,d x \] Input:
int((x + (x - 1)^(1/2))^(1/2),x)
Output:
int((x + (x - 1)^(1/2))^(1/2), x)
Time = 0.17 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.85 \[ \int \sqrt {\sqrt {-1+x}+x} \, dx=\frac {\sqrt {x -1}\, \sqrt {\sqrt {x -1}+x}}{6}+\frac {2 \sqrt {\sqrt {x -1}+x}\, x}{3}-\frac {\sqrt {\sqrt {x -1}+x}}{4}-\frac {3 \,\mathrm {log}\left (\frac {2 \sqrt {\sqrt {x -1}+x}+2 \sqrt {x -1}+1}{\sqrt {3}}\right )}{8} \] Input:
int(((x-1)^(1/2)+x)^(1/2),x)
Output:
(4*sqrt(x - 1)*sqrt(sqrt(x - 1) + x) + 16*sqrt(sqrt(x - 1) + x)*x - 6*sqrt (sqrt(x - 1) + x) - 9*log((2*sqrt(sqrt(x - 1) + x) + 2*sqrt(x - 1) + 1)/sq rt(3)))/24