I wonder if this integration can be calculated in a known form of any distribution. I need this in the context of prediction under the Bayesian modeling.
Question
$$ f(y) = \int N(y|\mu, \sigma^2) \cdot N(\mu| m, s^2) \cdot IG(\sigma^2| \alpha, \beta) d\mu d\sigma^2 $$ where $N(x|a, b^2)$ denotes a pdf of normal distribution with mean $a$ and variance $b^2$ and $IG(x|c, d)$ denotes that of the inverse gamma distribution with shape $c$ and scale $d$ (see this wiki page for the details of the inverse gamma distribution).
Trial
I first integrate $\mu$ out and get this result $$ \begin{array}{l} f(y) = {\Large \int}{\Large \int} N(y|\mu, \sigma^2) \cdot N(\mu| m, s^2) d\mu IG(\sigma^2| \alpha, \beta) d\sigma^2 \\ = {\Large \int} (\sigma^{-2} + s^{-2})^{-1/2} (\sigma^2)^{-\alpha -3/2} \\ \qquad \qquad \times \exp\Big\{ -\sigma^{-2}(y^2/2 + \beta) + \dfrac{1}{2} (\sigma^{-2} + s^{-2})^{-1}(\sigma^{-2}y +s^{-2}m)^2 \Big\}d\sigma^2 \end{array} $$ However, I'm stuck with the last integration.
Do you have any idea to complete this calculation? The integration in my question looks like an elementary example that may appear in the book "Introduction of the Bayesian Statistics", but to my shame, I have not found a solution yet.