Differential binomial

An expression of the type

$$x^m(a+bx^n)^p\,dx,$$

where $a$ and $b$ are real numbers, while $m$, $n$ and $p$ are rational numbers. The indefinite integral of a differential binomial,

$$\int x^m(a+bx^n)^p\,dx,$$

is reduced to an integral of rational functions if at least one of the numbers $p$, $(m+1)/n$ and $p+(m+1)/n$ is an integer. In all other cases, the integral of a differential binomial cannot be expressed by elementary functions (P.L. Chebyshev, 1853).

Comments

The statement on the reduction to an integral of rational functions is called the Chebyshev theorem on the integration of binomial differentials.