Computing the integrals of the form $\exp(P(x))$, $P(x)$ a polynomial

Hint: Assume $a<0$. We have

$$\int_{-\infty}^\infty e^{\large ax^4+bx^3+cx^2+dx+f}dx=e^f\int_{-\infty}^\infty\left(\sum_{n=0}^\infty \frac{(bx^3)^n}{n!}\right)\left(\sum_{m=0}^\infty\frac{(cx^2)^m}{m!}\right)\left(\sum_{p=0}^\infty \frac{(dx)^p}{p!}\right)e^{ax^4}dx $$

$$=\int_{-\infty}^\infty \sum_{n,m,p\ge0}^{n+m\equiv0(2)}\frac{b^nc^md^p}{n!m!p!}x^{3n+2m+p}e^{ax^4}dx $$

because the powers with $n+p\equiv1\bmod2$ contribute odd functions to the integrand, which vanish when integrated over the real line. From here, interchange summation and integration, use the substitution $u=x^4$, then dilate by $-a$ appropriately...

(So basically, the techniques in play here are: series expansion, self-cancellation of odd functions, interchange of limiting operations, and power/linear substitutions.)