Question
Let $F(x)$ be a cdf and $f(x)$ its pdf. Further let $h(x)$ be the pdf of some unrelated distribution. Further let $a,b,c \geq 1$ be constant numbers.
I have the following double integral: $$ A(r,s)=\int_s^\infty \int_0^{\frac{r-s}{a}} F(r-xb)f(r-xc)h(x)\,dx\,dr, $$ which I would like to compute for thousands of values of $r$ and $s$.
Thoughts
I can do this in a simple for loop, considering each integral as a new problem. Unfortunately this takes a very, very long time to compute. Therefore I would need a more efficient method like recognizing some type of convolution or cumulative integral (like computing $\int_{0}^s f(s-u) g(u) du, \int_0^s f(u) du$ for all $s$ can be done very quickly). To this end we may apply Fubini's theorem to find: $$ A(r,s) = \int_0^\infty \int_{ax+s}^\infty F(r-xb)f(r-xc)\,dr h(x)\, dx. $$ Which could be simplified significantly if $xb = xc$ but this is not the case..