Let $f \in C^1[0,\infty)$ be an increasing function with $f(0)>0$, suppose $\int_0^\infty \frac{1}{f(x)+f'(x)} < \infty$, prove that $\int_0^\infty \frac{1}{f(x)} < \infty$.

I find it weird since the behaviour of $f'$ is random, so I don't know how to control $f$ in terms of $f+f'$.