In Freedman and Proeyen's text on supergravity they derive the equation of motion for the gravitino using the second order formalism. However, I'm not exactly clear as to how they use partial integration and the identities given to manipulate it into the final form. They give the gravitino action as (9.1),
$$S_{3/2}=-\frac{1}{2\kappa^2}\int d^Dx~e\bar{\psi}_\mu\gamma^{\mu\nu\rho}D_\nu\psi_\rho,\tag{9.1}$$
and go on to say on p. 189,
"In the second order formalism, partial integration is valid, so it is sufficient to vary $\delta\bar{\psi}_\mu$ and multiply by 2 obtaining, $$\delta S_{3/2}=-\frac{1}{\kappa^2}\int d^Dx~e(D_\mu\bar{\epsilon})\gamma^{\mu\nu\rho}D_\nu\psi_\rho$$ $$=\frac{1}{\kappa^2}\int d^Dx~e\bar{\epsilon}\gamma^{\mu\nu\rho}D_\mu D_\nu\psi_\rho=\frac{1}{8\kappa^2}\int d^Dx~e\bar{\epsilon}\gamma^{\mu\nu\rho}R_{\mu\nu ab}\gamma^{ab}\psi_{\rho}.\tag{9.7}$$ We integrated by parts and used (8.37) to move to the second line$^4$ and then used the Ricci Identity (7.123) to obtain the last expression."
The relevant equation they mention in the text are:
$$\nabla_\mu\gamma_\nu=0,\tag{8.37}$$
and
$$[D_\mu,D_\nu]\Phi=\frac{1}{2}R_{\mu\nu ab}M^{ab}\Phi,$$
$$[D_\mu,D_\nu]V^a =R_{\mu\nu cb}\eta^{ac}V^b,$$
$$[D_\mu,D_\nu]\Psi=\frac{1}{4}R_{\mu\nu ab}\gamma^{ab}\Psi,\tag{7.123}$$
where the last equation is for a spinor. This is all in section 9.1 in the textbook.