# integration of differential binomial

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 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.

Title integration of differential binomial IntegrationOfDifferentialBinomial 2013-03-22 14:45:49 2013-03-22 14:45:49 rspuzio (6075) rspuzio (6075) 5 rspuzio (6075) Theorem msc 26A36