Evaluating a Function

Algebra Level 5

The function $f$ from the real numbers to the real numbers satisfies $f(1) = 4$ , and

$\begin{aligned} f(x+y) = & \left(1 + \frac {y}{x+1}\right) f(x) + \left(1 + \frac {x}{y+1} \right) f(y) \\ & + x^2y + xy + xy^2, \end{aligned}$

for $x, y \neq -1$ , $x, y$ real numbers. If $f \left( \frac {5}{3} \right)=\frac {a}{b}$ , where $a$ and $b$ are coprime positive integers, what is the value of $a+b$ ?