Integral of time function and time dependent arbitrary function with respect to time.

I don't understand your gamma attempt. Anyway, here goes mine. $\int_{A_0,0}^{y(T),T} e^t e^y \dot y dt = e^{T+y(T)}-e^{A_0} - \int_{A_0,0}^{y(T),T} e^t e^y dt$. Now, for that integral, try the new variable $x=e^t$, so $dx=e^tdt$ and let $\bar y(x) = y(t(x))=y(log(x))$, and substituting, we get $\int_{A_0,0}^{y(T),T} e^t e^y dt = \int_{A_0,1}^{\bar y(e^T),e^T} e^{\bar y(x)} dx$.

So, being able to (analitically) solve your integral seems to be equivalent to being able to solve this last one which, generically, is impossible (take $\bar y(x) = x^2$, for example).

For a numerical approach, many softwares could help you if you know the function y(x), but if you don't, that integral seems to strongly depend on the function $y(t)$.

Also, if you call $z(T)$ your integral, you get $\dot z = e^t e^y \dot y$, with the obvious initial conditions (becomes linear with $m=e^{y(t)}$. So if you happen to have an ODE for $y$, you could solve both simultaneously, or even more easily numerically.

I hope that helps. Where did you come across that integral?