Closed form of a SDE

Question

I have encountered a SDE : $$dS(t) = rS(t)(\alpha - S(t))dt + \sigma S(t)dW(t), S(0) = x.$$ I have tried to use $d(logS(t))$ to calculate the closed form solution, but in the end, I cannot find a way to cancel S(t). Anyone can share an idea how to use product rule to come up with a closed form solution? The closed form solution should be: $$S(t) = \frac{exp\{(r\alpha - \frac{1}{2}\sigma^2) t +\sigma W(t) \}}{\frac{1}{x}+r\int_{0}^{t}exp\{(r\alpha - \frac{1}{2}\sigma^2) s +\sigma W(s) \}ds}$$