  We construct a new type of quantum invariant of closed framed $3$-manifolds
with the vanishing first Betti number. The invariant is defined for any finite
dimensional Hopf algebra, such as small quantum groups, and is based on ideal
triangulations. We use the canonical element of the Heisenberg double, which
satisfies a pentagon equation, and graphical representations of $3$-manifolds
introduced by R. Benedetti and C. Petronio. The construction is simple and easy
to be understood intuitively; the pentagon equation reflects the Pachner
$(2,3)$ move of ideal triangulations and the non-involutiveness of the Hopf
algebra reflects framings. For an involutory Hopf algebra, the invariant
reduces to an invariant of closed combed $3$-manifolds. For an involutory
unimodular counimodular Hopf algebra, the invariant reduces to the topological
invariant of closed $3$-manifolds which is introduced in our previous paper. In
this paper we formalize the construction using more generally a Hopf monoid in
a symmetric pivotal category and use tensor networks for calculations.
