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In this paper, we discuss the inverse problem for a mixed Li\'enard typenonlinear oscillator equation $\ddot{x}+f(x)\dot{x}^2+g(x)\dot{x}+h(x)=0$,where $f(x),\,g(x)$ and $h(x)$ are arbitrary functions of $x$. Very recently,we have reported the Lie point symmetries of this equation. By exploiting theinterconnection between Jacobi last multiplier, Lie point symmetries andPrelle-Singer procedure we construct a time independent integral for the caseexhibiting maximal symmetry from which we identify the associated conservativenon-standard Lagrangian and Hamiltonian functions. The classical dynamics ofthe nonlinear oscillator is also discussed and certain special propertiesincluding isochronous oscillations are brought out.

The inverse problem of a mixed Li\'enard type nonlinear oscillator equation from symmetry perspective