Sreejato's triangle

Geometry Level 5

In triangle \(ABC\), \( AB = \sqrt{41} \) and \( AC = \sqrt{ 164} \). The internal angle bisector of \( BAC\) meets \(BC\) at \(D\). Given that lengths \(AD\) and \(DC\) are positive integers, what is the value of \(AD+DC\)?

This problem is proposed by Sreejato.