Features Integration of Differential Binomial

In other cases, the integral of the differential binomial (1), as proved by Chebyshev cannot be expressed through elementary functions (Chebyshev, 1947). In the first case the theorem after substitutions and small application of binomial Newton, the example reduces to integrating power function or fractional-rational function and no problems arise. After standard substitutions in the second and third cases and further transformations the presence of radicals of various degrees greatly complicates simplification element of integration, which causes a mistake in the process. Therefore, we can not only offer a methodological approach that avoids cumbersome transformations and in faster integration results in fractional rational function, but also give the proof for the general case. Consider the cases II and III.