Integrate square root of quotient polynomials I need to estimate this integral:
$$\displaystyle\int_a^\lambda \sqrt{\frac{(z-a)(z-b)}{z(z-1)}}dz$$ 
where $a$ and $b$ are closed to 1/2, and $z$ is complex.
Mathematica does not know how to do that. Any idea or reference ?

Not quite what you asked but using Mathematica 10.2 with the input:
FullSimplify[Integrate[Sqrt[(z - a) (z - b)/(z (z - 1))], z]]

I got the following output:
$$\left(\sqrt{\frac{(-a+z)(-b+z)}{(-1+z)z}}\left((a-b)(b-z)\sqrt{\frac{(-1+b)(a-z)}{(a-b)(-1+z)}}z\sqrt{\frac{b-z}{b-bz}}+(a-b)b\sqrt{\frac{(-1+a)(b-z)}{(-a+b)(-1+z)}}(-1+z)\sqrt{-\frac{(-1+b)(b-z)z}{b^2(-1+z)^2}}\text{EllipticE}\left[\text{ArcSin}\left[\sqrt{\frac{(-1+a)(b-z)}{(-a+b)(-1+z)}}\right],\frac{a-b}{(-1+a)b}\right]-(-1+b)\left((a-b)\sqrt{\frac{(-1+a)(b-z)}{(-a+b)(-1+z)}}(-1+z)\sqrt{-\frac{(-1+b)(b-z)z}{b^2(-1+z)^2}}\text{EllipticF}\left[\text{ArcSin}\left[\sqrt{\frac{(-1+a)(b-z)}{(-a+b)(-1 + z)}}\right],\frac{a-b}{(-1+a)b}\right]+2(-1+a)(b-z)\sqrt{\frac{(-1+b)z}{b(-1+z)}}\text{EllipticF}\left[\text{ArcSin}\left[\sqrt{\frac{b-z}{b-bz}}\right],\frac{(-1+a)b}{a-b}\right]+b(-1+a+b)(-1+z)\sqrt{-\frac{(-1+b)(b-z)z}{b^2(-1+z)^2}}\sqrt{\frac{b-z}{b-bz}}\text{EllipticPi}\left[b,\text{ArcSin}\left[\sqrt{\frac{b-z}{b-bz}}\right],\frac{(-1+a)b}{a-b}\right]\right)\right)\right)\bigg/\left((a-b)(b-z)\sqrt{\frac{(-1+b)(a-z)}{(a-b)(-1+z)}}\sqrt{\frac{b-z}{b-bz}}\right)$$
You could hopefully then manually apply your limits of integration.
