If $(x+a)^{100}=t_0+t_1+\cdots+t_{100}$ where $t_r=\dbinom{n}{r}x^{n-r}a^r$, then $\displaystyle \int\dfrac{2xdx}{\left (\displaystyle\sum_{r=0}^{50}(-1)^rt_{2r}\right )^2+\left (\displaystyle\sum_{r=0}^{49}(-1)^rt_{2r+1}\right )^2}=C+\dfrac{\alpha}{\beta(x^\eta+a^\eta)^\gamma},$

If $\beta\eta>0, GCD(|\alpha|,|\beta|)=1$, Calculate $\alpha+\beta+\eta+\gamma$