The quantum dynamics of carbon chains through H2 and He collisions in the interstellar medium (ISM) is an important step toward accurate modeling of their abundance in non-local thermodynamic equilibrium conditions. The C5(Σg+1) molecule is the longest pure carbon chain detected in the ISM to date. While He collisions are computationally easy to perform, the collision with much more abundant H2 is both complicated and computationally demanding. Using templates for approximating p-H2 collisional rates, such as scaling He rates and using a reduced 4D → 2D potential energy surface (PES), has limited applicability. On the other hand, any such approximation does not exist for o-H2. Therefore, a full rotational dynamics of C5 with both p- and o-H2 is performed considering both molecules as rigid-rotors. The PES is calculated using CCSD(T)-F12a/AVTZ, and a neural network fitting model has been carefully chosen to strictly obey spectroscopic accuracy and augment the PES. The augmented PES is then expanded into radial terms using the bispherical harmonics function, and close coupling calculations have been done to get the cross sections and, subsequently, rate coefficients for various rotational transitions of C5.