I have a cubic function $N_3(x) = a x^3 + b x^2 + c x + d$ which guaranteed is non-negative in each point on interval $x \in [0,1]$.
I building an other function $N_4(x) = \int_0^x{N_3(t)dt}$. Sure $N_4(x)$ is a fourth-degree polynomial function whose coefficients can be easily identified. It is easy to show that the function does not decrease on the whole interval. This implies that $N_4(x)$ is invertible.
Now let $y = N_4(x)$. The question is can we get an analytical form of expression $x = N_4^{-1}(y)$, where $N_4^{-1}$ is an inversion of $N_4$, i.e. $N_4^{-1}(N_4(\cdot))=I(\cdot)$?
EDIT: Seems I should solve a quartic equation $\frac{a}{4} x^4 + \frac{b}{3} x^3 + \frac{c}{2} x^2 + d x + y$ and get a dependence $x(y)$ while $x \in [0, 1]$.