Too much roots!

The rationalization of the expression

15+2+1\dfrac{1}{\sqrt{5} + \sqrt{2} + 1}

can be expressed as

A10B5+C2+DE,\dfrac{- A\sqrt{10} - B\sqrt{5} + C\sqrt{2} + D}{E},

where AA, BB, CC, DD, and EE are all positive integers.

Find the minimum value of A+B+C+D+EA + B + C + D + E.

This problem is part of the set "Xenophobia."


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