Analytical study of conformable fractional bohr hamiltonian with the extended form of the sextic potential

New analytical solutions of the conformable fractional Bohr Hamiltonian for triaxial nuclei have been derived, incorporating an extended form of the Sextic potential in the $\beta$ component of the collective nuclear potential, alongside a steep harmonic oscillator in the $\gamma$ component. This particular type of potential can specifically recover sextic, Davidson and harmonic potentials in particular cases. The energy spectra and corresponding wave functions are analytically determined through a novel conformable fractional extended Nikiforov-Uvarov method. Additionally, the derivation of the wave functions necessitates the application of the bi-confluent Heun functions. This study examines the relationship between the conformable fractional spectra of the potential and the $Z\left(5\right)$ benchmark spectrum. A systematic analysis is conducted to explore the spectral expansion as a function of both the fractional derivative order and the potential parameters. The normalized $B\left(E2\right)$ transition rates and corresponding energy spectra were computed and systematically compared with experimental measurements. The theoretical predictions demonstrate excellent agreement with empirical data across multiple isotopic chains, including ^112,114,116Pd, ^{120, 126, 128, 130, 132, 134}Xe, and ^192,194,196Pt.