$Version
(* 10.1.0  for Microsoft Windows (64-bit) (March 24, 2015) *)

Integrate[1/(Sqrt[x] Sqrt[1 - x + x^2]), {x, 1, 2}]
(* Integrate[1/(Sqrt[x] Sqrt[1 - x + x^2]), {x, 1, 2}] *)

Integrate[1/(Sqrt[x] Sqrt[1 - x + x^2]), x]
(* (2 (-1)^(1/6) Sqrt[1 - (-1)^(1/3)/x] Sqrt[1 + (-1)^(2/3)/x] x
     EllipticF[I ArcSinh[(-1)^(1/3)/Sqrt[x]], (-1)^(2/3)])/Sqrt[1 - x + x^2] *)

FullSimplify[(% /. x -> 2) - (% /. x -> 1)]
(* 2 (-1)^(1/6) (-EllipticF[I ArcSinh[(-1)^(1/3)], (-1)^(2/3)] +
     EllipticF[I ArcSinh[(-1)^(1/3)/Sqrt[2]], (-1)^(2/3)]) *)

% - NIntegrate[1/(Sqrt[x] Sqrt[1 - x + x^2]), {x, 1, 2}, WorkingPrecision -> 30]
(* 0. 10^-31 + 0. 10^-47 I *)

$Version

(* Out[1]= "8.0 for Microsoft Windows (64-bit) (October 7, 2011)" *)

f[x_] = 1/(Sqrt[x] Sqrt[1 - x + x^2]);

Timing[Integrate[1/(Sqrt[x] Sqrt[1 - x + x^2]), {x, 1, 2}] ]

(*
Out[3]= {3.011, 2 (-1)^(
  1/6) (-EllipticF[I ArcSinh[(-1)^(1/3)], (-1)^(2/3)] +
    EllipticF[I ArcSinh[(-1)^(1/3)/Sqrt[2]], (-1)^(2/3)])}
*)

% // N

(* Out[4]= {3.011, 0.646172 - 5.55112*10^-17 I} *)

NIntegrate[f[x], {x, 1, 2}]

(*
Out[5]= 0.646172
*)

Integrate[1/(Sqrt[x] Sqrt[1 - x + x^2]), x]

(*
Out[6]= (2 (-1)^(1/6) Sqrt[1 - (-1)^(1/3)/x] Sqrt[
 1 + (-1)^(2/3)/x] x EllipticF[I ArcSinh[(-1)^(1/3)/Sqrt[x]], (-1)^(
  2/3)])/Sqrt[1 - x + x^2]
*)

$Version

(* Out[2]=5.2 for Microsoft Windows x86 (64 bit) (June 20, 2005) *)

Timing[Integrate[1/(Sqrt[x] Sqrt[1 - x + x^2]), {x, 1, 2}] ]

(*
Out[4]=
{0.2030*Second, 2*(-1)^(1/6)*(-EllipticF[I*ArcSinh[(-1)^(1/3)], (-1)^(2/3)] +
    EllipticF[I*ArcSinh[(-1)^(1/3)/Sqrt[2]], (-1)^(2/3)])}
*)

N[InverseJacobiCN[-1/3, 3/4] - EllipticK[3/4], 20]
   0.64617199330515618196

NIntegrate[1/Sqrt[x (1 - x + x^2)], {x, 1, 2}, WorkingPrecision -> 20]
   0.64617199330515618293