I am trying to solve
$$\int_{x_0}^\infty\frac{\sqrt{a(b(x+c)^2+1)}}{(x-1)x^{3/2}}dx$$
where, $a,b,c,x_0,x\in\Bbb R$ and $b,c,x>0$ and $x_0>1$.
I tried a $\sinh$ substitution without too much success. I also tried something like $v=\sqrt{a(b(x+c)^2+1)}\quad$, but maybe I should first do some more transformations before attempting something like that.
This is not homework. I would be grateful for some hints.