# Differential binomial

An expression of the type

$$x^m(a+bx^n)^pdx,$$

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

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

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).