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!

The answer to your question is very simple. In fact the integral does not converge as $c>0$.

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.