AMC 12 2012b Problem 20

A trapezoid has side lengths $3$, $5$, $7$, and $11$. The sums of all the possible areas of the trapezoid can be written in the form of $r_1\sqrt{n_1}+r_2\sqrt{n_2}+r_3$, where $r_1$, $r_2$, and $r_3$ are rational numbers and $n_1$ and $n_2$ are positive integers not divisible by the square of any prime. What is the greatest integer less than or equal to $r_1+r_2+r_3+n_1+n_2$?

\textbf{(A)}\ 57 \textbf{(B)}\ 59 \textbf{(C)}\ 61 \textbf{(D)}\ 63 \textbf{(E)}\ 65