Tough Mordell equation $x^3+9=y^2$

Question

Solve the Mordell equation $x^3+9=y^2$ in integers. It's evident that $(-2,1), (0,3), (3,6), (6,15)$ are solutions, but is there a good way to find all solutions to this equation? I have already tried basic ideas from number theory, such as rewriting it as $(y−3)(y+3)=x^3$. Note that $\gcd(y−3,y+3)=1,2,3,6$. For $1$, I have verified there are no solution. For 2, I have gotten one solution, $(−2,1)$. The case for $3$ is quite difficult. If you consider an elliptic curve, I was looking at aspects of its rank, and how it may lead to a solution.