  We study the asymptotical behaviour of the moduli space of morphisms of given
anticanonical degree from a rational curve to a split toric variety, when the
degree goes to infinity. We obtain in this case a geometric analogue of Manin's
conjecture about rational points of bounded height on varieties defined over a
global field. The study is led through a generating series whose coefficients
lie in a Grothendieck ring of motives, the motivic height zeta function. In
order to establish convergence properties of this function, we use a notion of
eulerian motivic product. It relies on a construction of Denef and Loeser which
associates a virtual motive to a first order logic ring formula.
