There are general algorithms for finding all of the integer points on an elliptic curve given in Weietstrass form, but this special case is likely much easier. There’s a simple point of order 2, and the polynomial on the RHS generates a quadratic rather than a cubic number field (in fact it’s an imaginary quadratic field, since $D$ is positive). 


Still, I am unable to write down a complete algorithm for determining all integer points – maybe it’s still complicated, maybe I’m just not sufficiently observant. The ring of integers of $\mathbb{Q}\left({\sqrt{-D}}\right)$ may have a high class number, and there seem to be plenty of options for ideals dividing both $x$ and $x\pm\sqrt{-D}$. 


I’m interested in cases where $D$ has a small number of prime factors, but even the case where it is prime itself is interesting. 



Integer points on $y^2=x^3+Dx$, $D\in\mathbb{N}$ squarefree
I think that you're right that the problem is *easier* than the general case, but still it's not *easy*. I don't think there was an effective method known before Baker's work, and even now, you'll need to use linear forms in logs of some sort to get an effective upper bound of the form $|x|\le C(D)$. Even for $D$ prime, I doubt that you can get an effective bound for $C(D)$ in an elementary way. 


One point worth making is that if you try to use the 3-part of the class group of $\mathbb{Q}(\sqrt{-D})$, conjecturally you'll run into the problem that the 3-rank can be arbitrarily large. (I think that follows from the Cohen-Lenstra heuristics.) On the other hand, my recollection is that Noam Elkies had a heuristic argument suggesting that the number of integer solutions on your curve is bounded by a constant that does not depend on $D$. As further support for this heuristic, there's also an old theorem of mine that says that $\#E\_D(\mathbb Z)\le K^{1+\text{rank }E\_D(\mathbb Q)}$ for an absolute constant $K$, so recent conjectures that there is an upper bound for $\text{rank }E(\mathbb Q)$ for all elliptic curves $E/\mathbb Q$ would give uniformity of $\#E\_D(\mathbb Z)$. 


