This flexible pipe comprises a versatile and flexible tubular body (PP) which comprises, from the outside toward the inside, a pressure arch (1b-1c) with filaments (10) submerged in a layer of elastomer and wound in a helix with an angle of 54.7° in relation to the profile plates and an attached sealing layer (1a), as well as a pair of rigid end connector tips (EC) which comprises two coaxial cylindrical sleeves, an inner one (2) in which the sealing layer (1a) is fitted and held, and an outer one (7) which extends axially from the body side (PP) beyond the inner sleeve (2) while surrounding the transitional area, with variable diameter, of the pressure arch; the connector end is provided with a pair of flexible annular support elements (5, 6) with variable profile, arranged at this transitional area and inserted between the sealing layer (1a) and the pressure arch (1b-1c), and between the pressure arch (1b-1c) and the flange (7), respectively, avoiding shearing of the filaments upon inflation.A pipe of this type is intended in particular for the oil industry, for the transport of liquids or gases.