# Extended Gaussian integral

Is there a closed expression for the following definite integral?

$$\int_{- \infty}^\infty \exp \left(-\frac{ax^2}{2}+bx^3+cx^4\right) \, dx;$$

$a,b,c$ are constants.

I know that one can perform a series expansion in the terms with $b$ and $c$, but I think this integral can be expressed in Terms of Special functions. Is there a Special technique to do this?

• For $$b=0$$ and $$c=0$$, we have a simple Gaussian integral, if a is positive.
• For $$b=0$$ and $$c<0$$, we have two analytic expressions in terms of Bessel functions, depending on whether a is positive or negative.
• For $$b=0$$ and $$c>0$$, the integral diverges.
• For $$b\neq0$$ and $$c=0$$, the integral diverges.