deriving regular expressions from a regular language

Question

Given the language below, how do i find a regular expression for the language

L = {a ^n b ^m | n => 1, m =>1, nm =>3}

Answer 1

n>=1 and m>=1 and nm>=3 is true for each of the following:

n=1,m>=3

n>=3,m=1

n>=2,m>=2

So L = { abbb, abbbb, abbbbb, ... } U { aaab, aaaab, aaaaab, ... } U { a^n b^m | n>=2,m>=2 }

This regex should be equivalent to L:

((abbb(b*)) | (aaa(a*)b) | (aa(a*)bb(b*)))

There is probably a much more succinct answer than this.