Apart2[expr] partial fractions propagators of the form
1/[(q^2-m1^2)(q^2-m2^2)].
Overview, FAD, FeynAmpDenominator, ApartFF.
FAD[{q, m}, {q, M}, q - p]
Apart2[%]
StandardForm[FCE[%]]\frac{1}{\left(q^2-m^2\right).\left(q^2-M^2\right).(q-p)^2}
\frac{\frac{1}{\left(q^2-m^2\right).(q-p)^2}-\frac{1}{\left(q^2-M^2\right).(q-p)^2}}{m^2-M^2}
\frac{\text{FAD}[\{q,m\},-p+q]-\text{FAD}[\{q,M\},-p+q]}{m^2-M^2}
Apart2 can also handle Cartesian propagators with square roots. To disable this mode use text{Sqrt}to text{False}
int = CFAD[{{k, 0}, {+m^2, -1}, 1}, {{k - p, 0}, {0, -1}, 1}] GFAD[{{DE - Sqrt[CSPD[k, k]], 1}, 1}]
int // FeynAmpDenominatorCombine // Apart2\frac{1}{(\text{DE}-\sqrt{k^2}+i \eta ) (k^2+m^2-i \eta ).((k-p)^2-i \eta )}
\frac{\frac{\text{DE}}{(k^2+m^2-i \eta ).((k-p)^2-i \eta )}+\frac{1}{(\text{DE}-\sqrt{k^2}+i \eta ).((k-p)^2-i \eta )}+\frac{\sqrt{k^2}}{(k^2+m^2-i \eta ).((k-p)^2-i \eta )}}{\text{DE}^2+m^2}