In Algorithms for Computer Algebra in the last chapter about Risch algorithm, the Rothstein-Trager method is applied to see if an elementary function has an elementary integral. For this, the determinant of the Sylvester matrix is computed. If the resulting polynomial has constant coefficients, than the original function has an elementary integral.
Suppose a function (a+b)/c has an elementary integral. Does it follow that both a/c and b/c have elementary integrals?
No: I assume you mean that $a$, $b$ and $c$ are functions defined on a subset of the real numbers.
Take $a=x+f(x)$ and $b=x-f(x)$ and $c=x$ where $f(x)$ does not have an elementary integral/anti-derivative.
The converse is true by the linearity of the integral/anti-derivative.