Q:

Integral $\int_1 ^{h_1}\frac{1}{\sqrt{h}\sqrt{h^2-b}} dh$

I am stuck at the flowing analytical integration:
$\int_1 ^{h_1}\frac{1}{\sqrt{h}\sqrt{h^2-b}} dh$ where $0<b<1$ and $h_1>1$.
This can be recast as
 $C\int_1 ^{x_1} \frac{\sqrt{x}}{\left((1-b) x^3+ b\right)^{3/4}} dx$
if we assume $h=\sqrt{(1-b)x^3+b}$. 
C is some constant.
In Mathematica, I am getting Elliptic (upper one) or Hypergeometric ${}_2 F_1$ functions. I am not able to get those functions analytically by myself. I need your help to calculate those integrations (analytically). Mostly I need the answer for indefinite integration, then one can put the limits.
Thanks in advance. 

A:

Just put $h=\sqrt b\sec\phi$ in the indefinite integral to get
$$\int \frac{dh}{\sqrt h\sqrt {h^2-b}}=b^{-1/4}\int \sqrt{\sec\phi} d\phi$$
Next look here : Integral of root of $\sec x$

