Too much roots!

Algebra Level 4

The rationalization of the expression

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

can be expressed as

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

where $A$ , $B$ , $C$ , $D$ , and $E$ are all positive integers.

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

This problem is part of the set "Xenophobia."