This one is not that bad

Calculus Level 4

\[ \large \int_{\frac{\sqrt 2}2}^1 \dfrac1{x^5 \sqrt{4x^2-1}} \, dx \]

If the integral above can be expressed as \( -A + \dfrac {B\sqrt C}A + \dfrac \pi D \), where \(A\), \(B\), \(C\) and \(D\) are positive integers with \(A\) and \(B\) being coprime integers, and \(C\) square-free, find \(A+B+C+D\).

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