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We apply the averaging theory of first and second order for studying the limit cycles of generalized polynomial Linard systems of the form x ˙ = y - 1 ( x ) y ,     y ˙ = - x - f ( x ) - g ( x ) y - h ( x ) y 2 , \dot x = y - 1\left( x \right)y,\,\,\dot y =  - x - f\left( x \right) - g\left( x \right)y - h\left( x \right){y^2}, where  l ( x ) =  ∊l 1 ( x ) +  ∊ 2 l 2 ( x ),  f  ( x ) =  ∊ f 1 ( x ) +  ∊ 2 f 2 ( x ),  g ( x ) =  ∊g 1 ( x ) +  ∊ 2 g 2 ( x ) and  h ( x ) =  ∊h 1 ( x ) +  ∊ 2 h 2 ( x ) where  l k ( x ) has degree  m  and  f k ( x ),  g k ( x ) and  h k ( x ) have degree  n  for each  k  = 1, 2, and  ∊  is a small parameter.

Limit cycles of Liénard polynomial systems type by averaging method