How to solve these two integrals after substitution

Question

I am trying to solve this integral: $$ \int_{0}^\infty \frac{\exp\left[-\frac{1}{\beta(x_s^2-1)}(u^2+y_p-y_s)\right]\exp\left[-\frac{1}{\beta}(u-\sqrt{y_s})^2\right]}{\sqrt{\frac{\pi}{\beta(x_s^2-1)}(u^2+y_p-y_s)}}\,u\,du, $$ where $\beta,\,x_s,\,y_p$ and $y_s$ are constants and there is no relation between them. I change from $u$ to $t=\frac{u-\sqrt{y_s}}{\sqrt{\beta}}$ and I obtain these two integrals which are quite similar: $$ \int_{-\sqrt{\frac{y_s}{\beta}}}^\infty \frac{\exp\left[-\frac{1}{\beta(x_s^2-1)}(\beta(t+\sqrt{y_s})^2+y_p-y_s)-t^2\right]}{\sqrt{\frac{\pi}{\beta(x_s^2-1)}(\beta(t+\sqrt{y_s})^2+y_p-y_s)}}\,t \,dt $$ and $$ \int_{-\sqrt{\frac{y_s}{\beta}}}^\infty \frac{\exp\left[-\frac{1}{\beta(x_s^2-1)}(\beta(t+\sqrt{y_s})^2+y_p-y_s)-t^2\right]}{\sqrt{\frac{\pi}{\beta(x_s^2-1)}(\beta(t+\sqrt{y_s})^2+y_p-y_s)}}\, dt $$ but I don't know how to carry out the integration after performing the substitution. How can I compute the integrals? Thank you in advance.