integration of differential binomial

Theorem.  Let $a$, $b$, $c$, $\alpha$, $\beta$ be given real numbers and  $\alpha \beta \ne 0$.  The antiderivative

$$I=\int x^a{(\alpha +\beta x^b)}^c𝑑x$$

is expressible by 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

$$I=\frac{1}{b}{\alpha }^{\frac{a+1}{b}+c}{\beta }^{-\frac{a+1}{b}}{B}_y(\frac{1+a}{b},c-1)$$

$$=\frac{1}{1+a}{\alpha }^{\frac{a+1}{b}+c}{\beta }^{-\frac{a+1}{b}}{y}^{\frac{1+a}{b}}F(\frac{a+1}{b},2-c;\frac{1+a+b}{b};y)$$

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

Title	integration of differential binomial
Canonical name	IntegrationOfDifferentialBinomial
Date of creation	2013-03-22 14:45:49
Last modified on	2013-03-22 14:45:49
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	5
Author	rspuzio (6075)
Entry type	Theorem
Classification	msc 26A36