The solution is formed from the integral wrt s of the reciprical square root of (1/3)a s^3 + (1/2)b s^2 + c s + v0^2. The result depends strongly on the roots of the polynomial in s. If they are real the result will be an inverse Jacobi elliptic function equal to d t, d a constant. Inverting the result is usually something like
sin[y[s]] = sn[d t|m]
with y[s] not being too complicated. Mathematica doesn't handle this type of problem gracefully yet. See "Handbook of Elliptic Integrals For Engineers And Physicists", Bryd & Friedman, Springer-Verlag 1954.