Convert finite automata to regular expression
You have an error in your regular expression. The correct one is $(ab)^*ab c^*dd^*$ (notice that after reading $(ab)^*$ you are in state $A$ and that the edge from $B$ to $C$ is labelled $b$).
This can be rewritten as $(ab)^+ c^*d^+$ since $(ab)^*ab = (ab)^+$ and $dd^* = d^+$.
Another equivalent expressions is $a(ba)^*bc^*dd^*$ since $(ab)^+ = a(ba)^*b$.