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.


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