I came across the problem "find all integer solutions to $y^2=x^3+17$."

I've tried several things, without any success, and I was hoping that someone could help out. (Some ideas or a reference for where to find it are both appreciated)

By numerical calculation I have found that the following integer points $(x,y)$ lie on the curve

$(-1,4)$, $(-2,3)$, $(2,5)$, $(4,9)$, $(8,23)$, $(43,282)$, $(52,375)$, $(5234,378661)$ and this is probably all of them.

Thanks