What is the religious ruling regarding a husband returning to his wife after having sworn on the the Qur'an that he would not take her back, in order to appease his brother?

Divorcing one's wife merely due to a brother's insistence is not permissible, as divorce without necessity is disliked (makruh). The spouses should engage in dialogue and understanding, away from the interference of others. Swearing upon the Quran strengthens an oath, but whoever sees benefit in breaking it, let him break it and offer expiation for his oath. It is permissible to take back one's wife if this was not the third divorce. If she is still within her waiting period (iddah), you may take her back without a new contract, and it is recommended to have witnesses. If her waiting period has ended, a new contract is necessary.