Women and children were among the victims, security officials said.
In Karbala, a suicide bomber injured at least 10 people. The city is about 27 kilometers, or 17 miles, west of al-Musayab.
Karbala, a holy and historic city in Shiite Islam, is in Karbala province.
Amaq New Agency, the media voice for ISIS, claimed responsibility for the terror group in both attacks in separate statements Friday on social media.
Melted ice cream mixed with blood as the dead and injured lay on the ground, witnesses said.