Assumptions allowing to calculate an elliptic integral

There are two different issues:

1. suboptimal handling of elliptic integrals in Mathematica, this is why Integrate with appropriate assumptions doesn't provide satisfactory results.

2. unsatisfactory feedback of assumptions on the results

Before of playing with assumptions let's slightly reformulate the problem by changing the integration variable (x -> z == x/b): $$ int_{0}^{b} sqrt{ frac{(a-x)(b-x)}{x}} dx = b^{3/2} int_{0}^{1} sqrt{ frac{(frac{a}{b}-z)(1-z)}{z}} dz$$

Now the integral can be simply calculated with the appropriate assumption (a/b == c > 1):

b^(3/2) Integrate[ Sqrt[(c - x) (1 - x)/x], {x, 0, 1}, Assumptions -> c > 1]

(1/(3 (-1 + c))) b^(3/2) (2 Sqrt[-1 + c] (-1 + c^2) EllipticE[1/(1 - c)] 
  - 2 c (Sqrt[-1 + c] (1 + c) EllipticK[1/(1 - c)] 
  - 2 I (-1 + c) (EllipticK[1 - c] - I EllipticK[c])))

Ad.1 The result is not manifestly real ( see also this answer) however it can be easily checked by choosing various constants, e.g. in the OP we had a == 2 and b == 1, therefore

% /. c -> 2

1/3 (6 EllipticE[-1] - 4 (3 EllipticK[-1] - 2 I (EllipticK[-1] - I EllipticK[2])))

which is the same numerically as the exact integral

Chop @ N @ %

2.07216

Plot[ ReIm[(1/(3 (-1 + c))) (2 Sqrt[-1 + c] (-1 + c^2) EllipticE[
            1/(1 - c)] - 2 c (Sqrt[-1 + c] (1 + c) EllipticK[1/(1 - c)] 
            - 2 I (-1 + c) (EllipticK[1 - c] - I EllipticK[c])))], 
      {c, 0, 5}, Evaluated -> True, Exclusions -> c == 1, PlotStyle -> Thick]

This plot demonstrates that the integral is real for c > 1:

Mathematica 10 cannot simplify the result asssuming c > 1, one should play further with special functions or exploit capabilities of MathematicalFunctionData (new in version 11 ), which could help in providing manifestly real symbolic result.

Ad.2 The integral with a/b instead of c does not yield the result, evaluate e.g.

b^(3/2) Integrate[Sqrt[(a/b - x) (1 - x)/x], {x, 0, 1}, Assumptions -> a > b]

Since Artes has already pointed to my previous answer, and my opinion remains unchanged to this very day, let me supply the closed form for your integral, obtained using formula 233.07 in Byrd and Friedman:

With[{A = 2, B = 1},
     {NIntegrate[Sqrt[(A - x) (B - x)/x], {x, 0, B}, WorkingPrecision -> 20], 
      N[2/3 (A/B)^(3/2) ((1 + B/A) EllipticE[B/A] - (1 - B/A) EllipticK[B/A]), 20]}]
   {2.0721594194996321471, 2.0721594194996321443}

With (0 <= x <= b <= a) the result of

Integrate[(Sqrt[a - x]*Sqrt[b - x])/Sqrt[x], x]

can be transformed into the simple form

(2/3)*(Sqrt[a - x]*Sqrt[b - x]*Sqrt[x] + 
Sqrt[a]*(a + b)*EllipticE[ArcTan[Sqrt[x]/Sqrt[b - x]], b/a] - 
Sqrt[a]*(a - b)*EllipticF[ArcTan[Sqrt[x]/Sqrt[b - x]], b/a])

which can be verified by calculating the derivative w.r.t. x. The transformation involves a shift of Pi/2 of the argument and an expansion of the elliptic integrals of complex argument into real part and imaginary constant (disposable). Within the above range of parameters it may be plotted without difficulty:

a = 4; b = 2; Plot[{NIntegrate[(Sqrt[a - y]*Sqrt[b - y])/Sqrt[y], {y, 0, x}], 
(2/3)*(Sqrt[a - x]*Sqrt[b - x]*Sqrt[x] + 
Sqrt[a]*(a + b)*EllipticE[ArcTan[Sqrt[x]/Sqrt[b - x]], b/a] - 
Sqrt[a]*(a - b)*EllipticF[ArcTan[Sqrt[x]/Sqrt[b - x]], b/a])}, {x, 0, b}, 
PlotStyle -> {Blue, {Dashed, Orange}}]

The value at x=bcorresponds to the one obtained above with Byrd and Friedman.