I need to calculate the following integral:
$$\int_{-\infty}^{\infty}e^{a_1x^3+a_2x^2+a_3x+a_4y^3+a_5y^2+a_6y+a_7xy}\,dx\,dy$$
If there wasn't the cross term: $xy$ then it would be equal to the product of two Airy functions:
$$\int_{-\infty}^{\infty}e^{a_1x^3+a_2x^2+a_3x}\,dx\cdot\int_{-\infty}^{\infty}e^{a_4y^3+a_5y^2+a_6y}\,dy.$$
but I don't know how to reduce it to that form.