Simple Definite Integral 2

Calculus Level 4

\[\large{ I = \int_1^{16} \dfrac{\ln(x)}{\sqrt{x}(x+4)} \, {d}x}\]

If \(I\) can be expressed as \(A\ln(A) \tan^{-1} \left( \dfrac{B}{C} \right)\) where \(A,B,C\) are positive integers with \(\gcd(B,C) = 1\) and \(A\) is a prime number. Then find the value of \(A+B+C\).

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