For example if the derivative of $f(x)$ is$$f'(x)=\frac{(x-1)^2(x-3)}{(x-2)}$$ then $f(x)$ has a point of inflection at $x=1$.
But $f(x)$ is not a polynomial rational !
Is there a way to determine which antiderivatives are the ratio of to polynomials?
COMMENT$$x_0\space \text{is an inflection point of f}\iff f''(x_0)=0\space \text{and}\space f''(x_0-\epsilon)f''(x_0+\epsilon)\lt0$$ where $x_0+\epsilon$ and $x_0-\epsilon$ are in the neighborhood of $x_0$. However, you should keep in mind that a denominator that is not constant in a function, in general has an integral with logarithms or with another transcendent function such as trigonometers, for example $\int\dfrac{1}{1+x^2}dx=\arctan (x)+C$.
I thing the answer to your question is negative.