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."

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