= Solution
For the asymptotically free <beta function>, separation of variables gives
$$
\boxed{g(t)=\frac{g}{\sqrt{1+2bg^2t}}.}
$$
The anomalous-dimension integral is elementary:
$$
\log f(t)=2c\int_0^t\frac{g^2}{1+2bg^2u}\,du=\frac cb\log(1+2bg^2t),\qquad
\boxed{f(t)=(1+2bg^2t)^{c/b}\sim(2bg^2)^{c/b}t^{c/b}.}
$$
For a reference two-point factor regular and nonzero at the free coupling, $C(r,g(t))\to C(r,0)$, so the large-momentum correction is a power $c/b$ of $\log(p^2/\mu^2)$. These are <asymptotic freedom> and <logarithmic two-point scaling from a cubic beta function>. If the supplied beta and gamma expressions are leading small-coupling terms rather than exact functions, they fix the leading logarithmic exponent, while subleading corrections and the prefactor depend on higher orders.
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