Integral of exponent of a general polynomial Is there a general solution for integrals in the form:
$$\int_{-\infty}^{\infty}e^{\sum_{i=1}^{t} k_i x^{i}} dx$$
where $t$ is finite, a form of a solution that can also help is when the polynomial is denoted by $f(x)$, and the solution is expressed using it.
I could find solutions for the quadratic case but not for such a general case.
Thanks!
 A: Since you never ask for a closed form solution, how about this? Take the Taylor series of $\exp(x)$ up to some arbitrary degree:
$$\exp(x)=\sum_{i=0}^\infty \frac{x^i}{i!}$$
then compose the above expression with your polynomial $P(x)=\sum_j^t k_jx^j$
$$\exp(P(x))=\sum_{i=0}^\infty \frac{(\sum_j^t k_jx^j)^i}{i!}$$
You can use the multinomial theorem to expand $(\sum_j^t k_jx^j)^i$ to an expression that can be readily integrated. The series is infinite, but if you calculate up to a value for i that's large enough you should eventually get an approximation of some kind. The exact degree to which you calculate will depend on how quickly the series converges and how close you need the answer.
If you want to avoid choosing some arbitrary degree for a polynomial approximation, then I offer this: I've heard that it is at least possible to always rewrite the integral of $\exp\circ P$ as a generalized hypergeometric function, however I've only recently gotten on this subject and the exact method of doing so is still a little beyond me. Hopefully, the paper I link to should help you get on your way. In either case, I think your best bet will be to implement a generalized answer on a computer. Good luck!
