In the simulation of a physical process, I face the problem of computing $$S=\int_{\tau_1}^{\tau_2}\int_{z_1}^{z_2}\frac{e^{-\large{\frac{a z^2+b z+c}{\tau }}}}{\tau ^{3/2}}\,d\tau\,dz$$ (I leave the problem as it is set; for sure, completing the square and changing variables makes some expressions looking simpler but the problem remains the same). Concerning the conditions : $a>0$, $b>0$, $c>0$, $b^2-4ac \leq0$.
There is no problem computing one of the integrals since $$I=\int\frac{e^{-\large{\frac{a z^2+b z+c}{\tau }}}}{\tau ^{3/2}}\,dz=\frac{\sqrt{\pi }\, e^{\large{\frac{b^2-4 a c}{4 a \tau }}} \text{erf}\left(\large{\frac{2 a z+b}{2 \sqrt{a\tau} }}\right)}{2 \sqrt{a}\, \tau }$$ and $$J=\int\frac{e^{-\large{\frac{a z^2+b z+c}{\tau}}}}{\tau ^{3/2}}\,d\tau=-\frac{\sqrt{\pi } \,\text{erf}\left(\sqrt{\large{\frac{a z^2+b z+c}{\tau}}}\right)}{\sqrt{a z^2+b z+c}}$$ Beside numerical integration, I did not find any way to compute $\int I\,d\tau$ or $\int J\,dz$.
The only case for which I found an expression is if $b^2-4ac=0$. For such a case $$\int\frac{\sqrt{\pi }\, \text{erf}\left(\large{\frac{2 a z+b}{2 \sqrt{a\tau}} }\right)}{2 \sqrt{a}\, \tau }\,d\tau=-\frac{(2 a z+b) \, }{a \sqrt{\tau }} \, _2F_2\left(\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2};-\frac{(2 a z+b)^2}{4 a \tau }\right)$$
Would somebody come with some suggestions ?
Edit
It $\tau_1=0$ and $\tau_2=\infty$ the problem is quite simple since $$\int_{0}^{\infty}\int\frac{e^{-\large{\frac{a z^2+b z+c}{\tau}}}}{\tau ^{3/2}}\,d\tau\,dz=\frac{\sqrt{\pi } \log \left(2 \sqrt{a} \sqrt{a z^2+b z+c}+2 a z+b\right)}{\sqrt{a}}$$ but, unfortunately, $\tau_1$ and $\tau_2$ are just finite values.
