AMC 10 2021a Problem 13

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