Problematic third order term in saddle point method

Question

When evaluating $$I \equiv \int_{-\infty}^{\infty} dx\,e^{-f(x)}$$ using saddle point method we expand $f(x)$ as done here but my problem arises when I include higher order term for example$$f(x) \approx f(x_0) + \frac{1}{2}(x- x_0)^2 f''(x_0) + \frac{1}{6}(x- x_0)^3 f'''(x_0)+\cdots.$$ $$I \approx \int_{-\infty}^{+\infty} dx\, e^{-f(x_0) - \frac{1}{2}(x-x_0)^2 f''(x_0)-\frac{1}{6}(x- x_0)^3 f'''(x_0)}= e^{-f(x_0)}\int_{-\infty}^{\infty} dx\, e^{-\frac{1}{2}(x-x_0)^2 f''(x_0)-\frac{1}{6}(x- x_0)^3 f'''(x_0)}.$$ Howsoever $f'''(x_0)$ is small compared to $f''(x_0)$ but at very large $x$ it will be going to dominate for example following plot is for $e^{-x^2+.005x^3}$ Infact including higher order odd power term will result in blowing up the integral. What is going wrong here? Is it because we're doing asymptotic analysis so we're bound to get diverging term in the result?