Solving intergration equation

Question

I am trying to determine $\gamma$ such that $$\int_0^T\exp\left(\frac{3t(T-t/2)\gamma\sigma}{T^2}+\alpha t+\frac{\sigma^2}{2}\left[t-\frac{3t^2}{T^3}\left(T-\frac{t}{2}\right)^2\right]\right)\;dt=K,$$ for some constants $T$, $\sigma$, $\alpha$ and $K$. I think it can be solved analytically in terms of error function but I dont know how. I have tried bisection method using a trapezoidal method to numerically integrate but it fails.