Is there a simpler solution for finding the inverse of a cubic function?

[agricola]

p=ax^3+bX^2+cX is positive monotonic and has an inverse X=G(p)

But I can't invert it. Is there an analytic solution?
For given parameter values I can plot numeric values- it has a tilted S shape.

 

[HallsofIvy]

This is, by the way, a "cubic" equation, not "quartic". I take it, you mean by "is positive monotonic" that you are assuming values of a, b, and c such that the function is monotonic. Of course a cubic equation cannot always be positive. A true "quartic" (fourth degree) equation can but it cannot be monotonic.

There is a general formula for solutions to cubic equations that could, theoretically, give an inverse function for this, but it is extremely complicated.