Root of the problem

Algebra Level 4

\[ \large \frac{\sqrt{5-\sqrt{21}}}{\sqrt{32-2(5-\sqrt{21})} - \sqrt{21}} \]

If the value of the radical expression above equals to \(\sqrt{\dfrac{a}{b}} - \sqrt{\dfrac{c}{d}}\), where \(a,b,c\) and \(d\) are positive integers with \(\gcd(a,b) = \) \(\gcd(c,d) = 1\). Find \(a+b+c+d\).

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