How to integrate the following expression?

Question

I want to know what is the best way to integrate the following expression and get the simplest answer $$ \int_{u_-}^{u^+} \frac{1}{r_1r_2} \exp\left(\left[r_1^2+r_2^2-2u\right]^{1/2}+4su\right)du \tag{1} $$ I tried it by Mathematica and got $$ \frac{1}{16 s^{3/2}} \left( 4e^{\sqrt{r_1^2+r_2^2-2u}+4su}\sqrt{s} + e^{\frac{1}{8s}+2\left[r_1^2+r_2^2\right]s} \sqrt{2\pi}\ \mathrm{Erf}\left[\frac{1-4s\sqrt{r_1^2+r_2^2-2u}}{2\sqrt{2s}}\right] \right) \tag{2} $$ which must be calculated for $u_-$ and $u_+$. Is there any way to get a simpler answer, preferably without Error function?