Is $ \frac{x+y}{x^3 y^3 - x^3 y - x y^3 + 2 xy + 1} $ a formal group law? Is $ \frac{x+y}{x^3 y^3 - x^3 y - x y^3 + 2 xy + 1} $ a formal group law on the interval $[-1,1]$ ?
It is a lot of work to check on associativity imo.
Maybe there is a shortcut around checking associativity ? Or a way to check it faster ?
Is there an easy algoritm to check if a symmetric rational function $f(x,y)$ is a formal group law ?
 A: I remembered that sometimes mathematicians do work that goes beyond what naive manipulations can do (even with a computer), and so I decided to search online for rational formal group law, and found a lot of good information.
Your question was answered very well in the mid 70's by Robert Bismuth in Corollary 2.5.4 of his Masters thesis: The only rational (one-dimensional) formal group laws over a field are those of the form $\boxed{F(x,y)=\dfrac{x+y+axy}{1+bxy}}$. 
In particular, no rational functions with denominator of degree 3 are formal group laws.
That said, the proof in that thesis is long and somewhat hard to read. A much tidier proof for fields of characteristic zero is given in the 1991 paper "Rational formal group laws" by Robert Coleman and Francis Oisin McGuinness. More broadly, a (less explicit) characterization of algebraic formal group laws was done in Coleman's 1986 paper One-dimensional algebraic formal groups.
All of this is summarized and presented with more context in MSE's very own Alex Walker's excellent blog post, Formal Groups and Where to Find Them.
