Interesting function, it actually factors nicely as x(x+cis(pi/6))(x-cis(pi/6)(x+cis(-pi/6)(x-cis(-pi/6), where cis is a way to describe complex numbers.
~~This is a little bigbrain but one thing you could try is using the fact that the derivative of the inverse is the reciprocal of the derivative. Then the derivative of f-inverse is 1/(5x^4 + 3x^2 + 1), and we can find f-inverse by integrating via partial fraction decomp and using the initial condition that f(0)=0.~~
~~Plugging this into an integral calculator spits out an absolute monstrosity involving roots inside of logs and arctans, but it is doable because of the nice geometry of the derivative (quadratic in x^2).~~
Edit: Nevermind, this is NOT how that rule works.