2012 AMC 12B 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\qquad\textbf{(B)}\ 59\qquad\textbf{(C)}\ 61\qquad\textbf{(D)}\ 63\qquad\textbf{(E)}\ 65

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.