Integral of exponential function with polynomial argument I am looking for an effcient way to evaluate
$$
I = \int_{-\infty}^\infty dx\  e^{-ax+bx^2+cx^4}\\
\text{where } a,b,c \in \mathbb{R}^+
$$
I have already read about a solution involving the series expansion of the exponential here Computing the integrals of the form $\exp(P(x))$, $P(x)$ a polynomial but I am looking for something computationally more efficient...
Any help is greatly appreciated! 

One way to begin to attack this is to understand what the parameters $a$, $b$, and $c$ represent.  Obviously, $c<0$ for convergence.  But what if $c$ is very small in the following sense:
$$ I = \int_{-\infty}^{\infty} dx \: \exp{[-M (x^2 + a x) + c x^4]} $$
as $M \rightarrow \infty$?  Then you can apply Laplace's Method and expand the integral in an asymptotic expansion in $M$.  That is, a rescaling of the argument of the exponential can provide a simple approximation to the integral, in certain cases, where all you need at worst are Error functions and their derivatives.
In the most general case, you may try a simple numerical technique based on estimations of the extent of the exponential along the real axis.
