= Solution
This is the <Schwarzschild-de Sitter spacetime>. Its <geodesic Lagrangian> may be taken as
$$
\mathcal L=\frac12\left[-f\dot t^2+f^{-1}\dot r^2+r^2\dot\theta^2+r^2\sin^2\theta\,\dot\phi^2\right].
$$
Time-translation and rotational symmetry give three independent <integrals of motion>, which may be chosen as
$$
E=f\dot t,
\qquad
L_z=r^2\sin^2\theta\,\dot\phi,
\qquad
L^2=r^4\left(\dot\theta^2+\sin^2\theta\,\dot\phi^2\right).
$$
The constant norm of the tangent is $g_{ab}\dot x^a\dot x^b=-\kappa$, where $\kappa=1$ for a timelike geodesic parametrized by proper time and $\kappa=0$ for a <null geodesic>. Substitution gives
$$
-\kappa=-\frac{E^2}{f}+\frac{\dot r^2}{f}+\frac{L^2}{r^2},
$$
and hence
$$
\boxed{\dot r^2+V(r)=E^2,
\qquad
V(r)=f(r)\left(\kappa+\frac{L^2}{r^2}\right).}
$$
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