\(f(x; \mu, \alpha) = \sqrt{\frac{1}{2{\pi}{\alpha}x^3}}exp\frac{(x-µ)^2}{2{\alpha}{\mu}^2x}\)

I want to convert it to the exponential form below or as close as possible:

\(f(x; \mu; \alpha) = exp(\frac{x\mu - b(\mu)}{\alpha} + c(y, \alpha))\)

I get to this point and then am not sure how to reach a form that is closer to matching the preferred form:

\(f(x; \mu, \alpha) = exp(-log (\frac{1}{2{\pi}{\alpha}x^3}) - \frac{x^2 + 2x{\mu} - {\mu}^2}{2{\alpha}{\mu}^2x})\)

\(f(x; \mu, \alpha) = exp(-\frac{x^2 +2x\mu - \mu^2}{2\alpha\mu^2x} + log(\frac{1}{2\pi\alpha x^3}))\)

Ideally I would want to end up with something along the lines of this, but I'm just not sure how to get there :

\(f(x; \mu, \alpha) = exp(- \frac{x\mu - x}{\alpha^2} + log(\frac{1}{2\pi\alpha x^3}))\)

Any help would be greatly appreciated. This isn't my area of strength so I may have made one or more silly mistakes along the way but I'm happy to have them pointed out!

Thanks