An exponential of a quadratic function where the first term has negative coefficient is a normal distribution. In particular, any function of the following form, where $c=b^2/(4a)+\ln(-a/\pi)/2$ and $a$ is negative, $$f(x)=e^{ax^2+bx+c}$$ can be turned into the pdf of a normal distribution function. That means the integral of this function can be written in terms of a standard normal cdf. For further discussion see: http://en.wikipedia.org/wiki/Normal_distribution#General_normal_distribution
This is extremely useful for quickly approximating integrals of functions who take this form, since normal cdf tables exist in every programming language, and these integrals are often slow to numerically compute.
I'm looking for a similar idea when $a$ is positive. My integral is specifically between $\alpha$ and $\beta$ where $0\leq\alpha<\beta\leq1$. Numerical integration is proving too slow to be useful. I have also tried turning the integral into multiple forms of infinite sums via Taylor series, but none of these forms have been quickly computable. Thanks!