What is the volume of tetrahedron $ABCD$ with edge lengths $AB = 2$, $AC = 3$, $AD = 4$, $BC = \sqrt{13}$, $BD = 2\sqrt{5}$, and $CD = 5$ ? $\textbf{(A)} ~3 \qquad$ $\textbf{(B)} ~2\sqrt{3} \qquad$ $\textbf{(C)} ~4\qquad$ $\textbf{(D)} ~3\sqrt{3}\qquad$ $\textbf{(E)} ~6$