  We consider a cosmological model in a
Friedmann--Lema\^{\i}tre--Robertson--Walker background space with an ideal gas
defined in Weyl Integrable gravity. In the Einstein-Weyl theory a scalar field
is introduced in a geometric way. Furthermore, the scalar field and the ideal
gas interact in the gravitational Action Integral. Furthermore, we introduce a
potential term for the scalar field potential and we show that the field
equations admit a minisuperspace description. Noether's theorem is applied for
the constraint of the potential function and the corresponding conservation
laws are constructed. Finally, we solve the Hamilton-Jacobi equation for the
cosmological model and we derive a family of new solutions in Weyl Integrable
cosmology. Some closed-form expressions for the Hubble function are presented.
