We report ab initio (LDA + U-sc) calculations of thermoelastic properties of ferric iron (Fe3+)- and aluminum (Al)-bearing bridgmanite (MgSiO3 perovskite), the main Earth forming phase, at relevant pressure and temperature conditions and compositions. Three coupled substitutions, namely, [Al](Mg)-[Al](Si), [Fe3+](Mg)-[Fe3+](Si), and [Fe3+](Mg)-[Al](Si) have been investigated. Aggregate elastic moduli and sound velocities are successfully compared with limited experimental data available. In the case of [Fe3+](Mg)-[Fe3+](Si) substitution, the high-spin (S = 5/2) to low-spin (S = 1/2) crossover in [Fe3+](Si) induces a volume collapse and elastic anomalies across the transition region. However, the associated anomalies should disappear in the presence of aluminum in the most favorable substitution, i.e., [Fe3+](Mg)-[Al](Si). Calculated elastic properties along a lower mantle model geotherm suggest that the elastic behavior of bridgmanite with simultaneous substitution of Fe2O3 and Al2O3 in equal proportions or with Al2O3 in excess should be similar to that of (Mg,Fe2+)SiO3 bridgmanite. However, excess of Fe2O3 should produce elastic anomalies in the crossover pressure region.