The OP really seeks the distributions with log-concave density functions -the quotient he differentiates is the reciprocal of the one we examine in order to determine the log-concavity or log-convexity of a function.
Specifically: For $f$ to be log-concave, it means that $\ln f$ is concave, for which we require that
$$\frac {d^2}{dx^2} \ln f \leq 0 \implies \frac {d}{dx}\left (\frac {f'}{f}\right) \leq 0 \implies f''\cdot f-(f')^2 \leq 0$$
From his part, the OP wants
$$\frac {d}{dx} \left (\frac {F'}{F''}\right) \geq 0 \implies \frac {d}{dx} \left (\frac {f}{f'}\right) \geq 0 \implies (f')^2 - f\cdot f'' \geq 0 $$
and upon re-arranging, the OP wants
$$ f\cdot f'' - (f')^2 \leq 0$$
which is the condition for log-concavity, not log-convexity. It is the useexpression of the condition onthrough the use of the reciprocal quotient that may cause some confusion.
A good free resource on some of the "named" distributions that have log-concave densities is
Mark Bagnoli and Ted Bergstrom. "Log-concave Probability and its Applications" 2004
It focuses on log-concavity but it also contains results on log-convexity. We see that the OP's assertion that "Weibull, normal, log-normal, exponential, gamma" have log-concave densities is not fully correct: for example, the normal and the exponential distribution do have log-concave densities, while Weibull and gamma have also log-concave densities for some of their incarnations, while for others they have log-convex densities. The log-normal has a density that is neither log-concave, nor log-convex.
Another resource that examines log-concavity and log-convexity more abstractly and rigorously is