A calculus problem by Ben Habeahan

Calculus Level 4

\[\frac {1+1}{1^2{(1+2)}^2} +\frac {2+1}{2^2{(2+2)}^2} +\frac {3+1}{3^2{(3+2)}^2} + \ldots = \frac{A}{B}\]

Find \( A+B \) where \(A\) and \(B\) are coprime positive integers.