Because the angles are in an arithmetic progression, and the angles add up to $180^{\circ}$, the second largest angle in the triangle must be $60^{\circ}$. Also, the side opposite of that angle must be the second longest because of the angle-side relationship. Any of the three sides, $4$, $5$, or $x$, could be the second longest side of the triangle.
The law of cosines can be applied to solve for $x$ in all three cases.
When the second longest side is $5$, we get that $5^2 = 4^2 + x^2 - 2(4)(x)\cos 60^{\circ}$, therefore $x^2 - 4x - 9 = 0$. By using the quadratic formula,
$x = \frac {4 + \sqrt{16 + 36}}{2}$, therefore $x = 2 + \sqrt{13}$.
When the second longest side is $x$, we get that $x^2 = 5^2 + 4^2 - 40\cos 60^{\circ}$, therefore $x = \sqrt{21}$.
When the second longest side is $4$, we get that $4^2 = 5^2 + x^2 - 2(5)(x)\cos 60^{\circ}$, therefore $x^2 - 5x + 9 = 0$. Using the quadratic formula,
$x = \frac {5 + \sqrt{25 - 36}}{2}$. However, $\sqrt{-11}$ is not real, therefore the second longest side cannot equal $4$.
Adding the two other possibilities gets $2 + \sqrt{13} + \sqrt{21}$, with $a = 2, b=13$, and $c=21$. $a + b + c = 36$, which is answer choice $\text{(A)}$.