2017 Cayley Problem 23

The triangle with side lengths 6, 8 and 10 is right-angled, while the triangle with side lengths 6, 8 and 9 is an acute triangle and the triangle with side lengths 6, 8 and 11 is an obtuse triangle. An obtuse triangle with positive area has side lengths 10, 17 and x. If x is an integer, what is the sum of all possible values of x?

\textbf{(A)}\ 161\quad \textbf{(B)}\ 198\quad \textbf{(C)}\ 63\quad \textbf{(D)}\ 323\quad \textbf{(E)}\ 224

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Cayley contests here.