Considering: $f(x) = \frac{1}{\sigma_x\sqrt{2\pi}}e^{-\frac{1}{2}(\frac{x}{\sigma_x})^2}$
$g_i(x) = \frac{1}{\sigma_i\sqrt{2\pi}}e^{-\frac{1}{2}(\frac{a_i+b_ix}{\sigma_i})^2}$
Is there a closed-form expression for this integral?$\int_{-\infty}^{+\infty} \left(f(x)\cdot\prod_i g_i(x)\right) \, \mathrm{d} x$