MATHEMATICA or MAPLE do not work?

As Robert noted due to a different exponential your inequality is worse than known for large x. It makes sense for x near zero. But for such values it is more natural to seek for improvements in terms of powers of x, not exponentials. But on this field it seems impossible to compete with known excellent inequalities via continuous fractions and Pade approximants.

It seems possible to evaluate the integral by the long calculation say in terms of the Meier G--function, but what for? What is your motivation?

Note the correct name - Abramowitz Milton.

And also 5 volumes of Prudnikov, Brychkov, Marichev - Integrals and Series. They are not only a source for tables and formulas but also a concise source of sp. func. identities and other properties.

Andrews G.E., Askey R., Roy R. Special functions. А very good book, unofficial continuation of Harry Bateman books.

But exactly in your case it seems a problem of convergence is not so important as the Gauss functions are all just polynomes.

Thank you. But incomplete beta is not good for complex parameters, with branch cuts and so on. So it seems difficult to use the formulae practically, besides some other symbols for initial integral.

Thank you Carlo, and to MATEMATICA. Will try to use it, simplify.