Analytic expression for implicit function

Question

I have a function defined as

f[k_]:= Maximize[{(x^2/(1+x^2))((1-x^(2k))/(1+x^(2k+1)))^2, 0 <= x <= 1}, x]

I would like to find an analytic form of $f$ as a function of $k$.

Differentiating the argument of the Maximize in $f$ with respect to $x$, I obtain the condition:

1+x^(3+4k) - x^(2k)(1+x)(x^2(1+2k)- x + (1+2k)) == 0

Is there a way to get the analytic form of $f$ as a function of $k$ alone?

Answer 1

Suprisingly, Maxima does find an explicit formula for the zeros of that polynomial. The expressions are truly scary, however... and you'd need to check if your solutions are at the ends of the interval, not where the derivative is zero (at first glance, the solutions seem to be complex, not real).

What do you need the expressions for? Perhaps a numerical solution of the resulting equation is more practical.