Holomorphic embedding (HE) method has gained increasing attention in power system analysis because of its robustness. This paper further extends the HE theories into the power flow calculation in hybrid AC/DC active distribution networks (ADNs). A flexible holomorphic embedding solution (FHES) method is proposed for the power flow calculation in AC/DC ADNs in the grid-connected and islanded modes. This paper also develops the HE models of voltage source converters (VSCs) with active and reactive power control modes. The HE models of loads in AC/DC ADNs are also formulated by taking full account of static voltage and frequency characteristics. Furthermore, the HE models of the distributed generators (DGs) under various control modes are developed. A sequential iterative method is employed to solve the developed HE models of AC/DC ADNs in the grid-connected and islanded modes. The performance of the proposed FHES method is evaluated using the modified IEEE 33-node and IEEE 123-node AC/DC ADNs. Finally, the testing results validate the accuracy and robustness of the proposed FHES method.