Early Lutheran theologians were against divorce, but apparently thought that the Bible did not explicitly forbid polygamy.
Luther said that monogamy is the standard under normal circuмstances, but extreme exceptions could exist. I can't find a reference, but I believe Luther gave as an example, a wife with leprosy. He apparently felt, among evil things, bigamy was better than divorce or adultery. He also said the state could not legalise polygamy in general.
I confess that I cannot forbid a person to marry several wives, for it does not contradict the Scripture. If a man wishes to marry more than one wife he should be asked whether he is satisfied in his conscience that he may do so in accordance with the word of God. In such a case the civil authority has nothing to do in the matter.
Philip Melanchthon said that Henry VIII had no need to risk schism to divorce his first wife, but could have married another.
Philip of Hesse married Christine of Saxony in 1523. He was 19 and she was a few days short of 18 at the time. Philip committed adultery within a "few weeks". (That sounds to me like he was unfit for marriage.)
After years of Philip living "constantly in a state of adultery and fornication" (per Luther), and with some degree of approval from Lutheran theologians and from his wife, he took a second "wife" in 1540. It is said that Philip found Christine ugly, but he had 7 children with her by 1540 and would have 3 more before she died in 1549. He had 9 children with his second "wife", 6 while Christine was alive.
Luther allegedly gave his advice in the confessional and did not formally acknowledge he had approved the 1540 marriage.