I am solving the following problem in heat transfer using the Laplace transform
$$\rho\,c{\frac {\partial }{\partial t}}T \left( x,t \right) =k{\frac { \partial ^{2}}{\partial {x}^{2}}}T \left( x,t \right) +q\kappa\,{ {\rm e}^{\kappa\,x}} $$
in the domain $-\infty < x < 0$ and with specified initial and boundary conditions.
When the inverse laplace transform is applied the following integral appears:
$$\int _{0}^{t}\!{{\rm e}^{-{\frac {\rho\,c{x}^{2}}{4k\tau}}+{\frac {{\kappa}^{2}kt}{\rho\,c}}-{\frac {{\kappa}^{2}k\tau}{\rho\,c}}}} \left( -2\,{\tau}^{-3/2}+{\frac {\rho\,c{x}^{2}}{{\tau}^{5/2}k}} \right) {d\tau} $$
I think that such integral can be solved in terms of the error function and its relatives. Please let me know what is the possible closed form of such integral. Many thanks.