Second order differential equation with function of Jacobi elliptic function as a solution

Question

Many years ago, I remember reading in a book how to transform the following ODE $$ f'' = af^3 -bf + c/f^3 $$ into a form that lead to a solution in terms of Jacobi elliptic functions. If I am not mistaken, the final solution was of the form $$ f = \sqrt{\alpha + \beta\, {\rm sn}^2} $$ where sn is the Jacobi elliptic sine function. (This might have been in a limit where sn becomes $\tanh$.) However, I do not see how to prove this.