I'm interested in determining all (or as many as possible) members of the class of real functions $T(x)$ such that the definite integral
$$I = \int_{dom(T)} \exp(T(x))dx $$ over all values where $T$ is real has an elementary expression (basic operations, exponentials, logarithms, factorials, maybe even some more well-known special functions if need be, but nothing involving hypergeometrics etc.)
I'm familiar with the result that a function $T(x)$ has an $indefinite$ integral iff there's a rational solution $h$ to the equation $1 = h' + hT'$, but that's not what I'm looking for here: for instance $T(x) = \ln(x)$ has a trivial indefinite integral, $\int \exp(\ln(x))dx = \int x dx = \frac12x^2+C$, but the definite integral $\int_0^\infty xdx \to \infty$ diverges, so it doesn't work. By contrast, the choice of $T(x) = -x^2$ is well-known to have no indefinite integral, but $\int_{-\infty}^\infty \exp(-x^2)dx = \sqrt{\pi}$, so it satisfies my criterion.
Is this even a decidable problem? Clearly we can identify certain functions that satisfy this criterion- for instance, any quadratic with a negative coefficient on the $x^2$ term, or anything of the form $a x + b \ln x$ if $a<0$. Is there some way to enumerate more of them, beyond just guessing and checking in a CAS? There's quite a few that seem counterintuitive, for instance $T(x) = ax - be^x$ doesn't have an indefinite integral at all, but does have $I = \Gamma(a)/b^a$.
Thanks in advance!