Is there a closed-form expression for the integral of this product of gaussian functions?

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?$$\int_{-\infty}^{+\infty} \left(f(x)\cdot\prod_i g_i(x)\right) \, \mathrm{d} x$$

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Are you aware that the product of Gaussians is again a Gaussian? – joriki Nov 28 '12 at 13:40
Sorry, I'm learning how this works. I'm aware of it. – Medical physicist Nov 28 '12 at 13:50
Then it would help if you explained where you see an obstacle to combining the Gaussians into a single Gaussian and integrating it. – joriki Nov 28 '12 at 13:51
When $i=1$, I obtain $$\frac{1}{\sqrt{2\pi((b\sigma_x)^2+\sigma_i^2)}}e^{-\frac{1}{2}(\frac{a^2}{(b\sig‌​ma_x)^2+\sigma_i^2})}$$ I would expect something similar for any $i$. – Medical physicist Nov 28 '12 at 13:58
Isn't the question general enough leaving out $f(x)$ and throwing that as a special case of the $g_i(x)$, with $a_i=0$, $b_i=1$, and $\sigma_i=\sigma_x$? So you're really just asking whether there's a closed form for the product of Gaussians with different parameters. See this article for the case where you've only got two: google.com/… – Gwyn W Nov 28 '12 at 14:00

Nice! I suppose that the next step is to try to convolve it to obtain a normal distribution a la $i=1$. – Medical physicist Nov 28 '12 at 14:46
@Medicalphysicist : The exponential of $-1/2$ times the function above is a scalar multiple of a normal density function centered at $-B/(2A)$, with variance $1/A$. – Michael Hardy Nov 28 '12 at 17:34