  In this paper, we consider a second order nonlinear ordinary differential
equation of the form
  $\ddot{x}+k_1\frac{\dot{x}^2}{x}+(k_2+k_3x)\dot{x}+k_4x^3+k_5x^2+k_6x=0$,
where $k_i$'s, $i=1,2,...,6,$ are arbitrary parameters. By using the modified
Prelle-Singer procedure, we identify five new integrable cases in this equation
besides two known integrable cases, namely (i) $k_2=0, k_3=0$ and (ii) $k_1=0,
k_2=0, k_5=0$. Among these five, four equations admit time dependent first
integrals and the remaining one admits time independent first integral. From
the time independent first integral, nonstandard Hamiltonian structure is
deduced thereby proving the Liouville sense of integrability. In the case of
time dependent integrals, we either explicitly integrate the system or
transform to a time-independent case and deduce the underlying Hamiltonian
structure. We also demonstrate that the above second order ordinary
differential equation is intimately related to the two-dimensional
Lotka-Volterra (LV) system. From the integrable parameters of above nonlinear
equation and all the known integrable cases of the latter can be deduced
thereby.
