Automatic multi-classification of breast cancer histopathological images has remained one of the top-priority research areas in the field of biomedical informatics, due to the great clinical significance of multi-classification in providing diagnosis and prognosis of breast cancer. In this work, two machine learning approaches are thoroughly explored and compared for the task of automatic magnification-dependent multi-classification on a balanced BreakHis dataset for the detection of breast cancer. The first approach is based on handcrafted features which are extracted using Hu moment, color histogram, and Haralick textures. The extracted features are then utilized to train the conventional classifiers, while the second approach is based on transfer learning where the pre-existing networks (VGG16, VGG19, and ResNet50) are utilized as feature extractor and as a baseline model. The results reveal that the use of pre-trained networks as feature extractor exhibited superior performance in contrast to baseline approach and handcrafted approach for all the magnifications. Moreover, it has been observed that the augmentation plays a pivotal role in further enhancing the classification accuracy. In this context, the VGG16 network with linear SVM provides the highest accuracy that is computed in two forms, (a) patch-based accuracies (93.97% for 40×, 92.92% for 100×, 91.23% for 200×, and 91.79% for 400×); (b) patient-based accuracies (93.25% for 40×, 91.87% for 100×, 91.5% for 200×, and 92.31% for 400×) for the classification of magnification-dependent histopathological images. Additionally, "Fibro-adenoma" (benign) and "Mucous Carcinoma" (malignant) classes have been found to be the most complex classes for the entire magnification factors.