differential equations - Criterion for finite time blowup of an ODE

(This used to be a comment, but I think it deserves to be an answer, after mulling over it a bit.)

I don't think your criteria are quite correct. Some counterexamples:

Let $f(x) = - x^2$, and $x(0) = -1$. This ODE blows up in finite time toward $-infty$. But $int_{-1}^infty dx / f(x) $ diverges due to the singularity at $x = 0$.

Similarly, for any $f(x)geq 0$ such that $f(0) = 0$, for any initial value $x(0) < 0$ we must have $x(0) leq x(t) leq 0$ for any $t > 0$. Hence we have global existence (no blow up). But if we take $f(x) = sqrt{|x|}$ if $|x| leq 1$ and $f(x) = x^2$ if $|x| > 1$, then $1/f(x)$ is integrable, and in particular $int_{x_0}^infty dx/f(x) < infty$ for any $x_0$.

One key point used in your examples $f(x) = x^2$ and $f(x) = sqrt{x}$ with initial data positive, is that there are no stationary points. And so for any positive initial datum the evolution eventually goes toward $xto infty$, and so the question of blowup reduces to a question of how fast that happens. And for that the integral test is a good one.

Another way of saying this is that you wanted to use the equality

$$ int_0^s dt = int_{x(0)}^{x(s)} frac{dx}{f(x)} $$

but you falsely presupposed that the end state is necessarily $x(s) = infty$.