Non linear SDE population Dynamics

Question

I have a simple stochastic Ito-diffusion process governing $X(t)$ where $$dX(t)=aX(1-X)+bX(1-X)dW(t)$$. I do not need to solve this SDE, but i would like to say something about the expected value of $X(t)$ as a function of $a$ and $b$. I know that i can look at $$S(x)=\int_{a}^{x}\exp \left(-\int_{a}^{z}\frac{2ay(1-y)}{(by(1-y))^2}dy\right)dz$$ to get some idea about asymptotic values of $X(t)$. The fokker plank equation associated with the SDE also did not help me go further. I tried the lamperti transformation with $Z(t)=\frac{\log(\frac{x}{1-x})}{b}$ with $$dZ(t)=\left(\frac{a}{b}-\frac{b(1-2x)}{2}\right)dt+dW_1(t)$$ where $W_1$ is a standard wiener processes. Unfortunately, this did not get me very far either. Is there someother trick that i can use to get an expression for $\mathbb{E}X(t)$ for finite $t$