Theorem. Let $a$ , $b$ , $c$ , $\alpha$ , $\beta$ be given real numbers and $\alpha\beta \neq 0$ . The antiderivative $$I = \int x^a(\alpha+\beta x^b)^c\,dx$$ is expressible by means of the elementary functions only in the three cases: $(1)\,\, \frac{a+1}{b}+c\in\mathbb{Z}$ , $(2)\,\, \frac{a+1}{b}\in\mathbb{Z}$ , $(3)\,\, c\in\mathbb{Z}$

In accordance with P. L. Chebyshev (1821$-$ 1894), who has proven this theorem, the expression $x^a(\alpha+\beta x^b)^c\,dx$ is called a differential binomial.

It may be worth noting that the differential binomial may be expressed in terms of the incomplete beta function and the hypergeometric function. Define $y = \beta x^b / \alpha$ . Then we have $$I = {1 \over b} \alpha^{{a + 1 \over b} + c} \beta^{-{a + 1 \over b}} B_y \left( {1 + a \over b}, c - 1 \right)$$ $$= {1 \over 1 + a} \alpha^{{a + 1 \over b} + c} \beta^{-{a + 1 \over b}} y^{1 + a \over b} F \left( {a + 1 \over b}, 2-c; {1 + a + b \over b}; y \right)$$

Chebyshev's theorem then follows from the theorem on elementary cases of the hypergeometric function.