AMC 12 2013a Problem 12

The angles in a particular triangle are in arithmetic progression, and the side lengths are $4,5,x$. The sum of the possible values of $x$ equals $a+\sqrt{b}+\sqrt{c}$ where $a, b$, and $c$ are positive integers. What is $a+b+c$?

\textbf{(A)}\ 36 \textbf{(B)}\ 38 \textbf{(C)}\ 40 \textbf{(D)}\ 42 \textbf{(E)}\ 44