Table of Integrals, Series, and Products, 8th Edition

2.2 Algebraic functions

2.20 Introduction

2.201 The integrals ${\displaystyle \int R\left(x\text{,}{\left(\frac{\text{α}x+\text{β}}{\text{γ}x+\text{δ}}\right)}^r,{\left(\frac{\text{α}x+\text{β}}{\text{γ}x+\text{δ}}\right)}^s\text{,}\dots \right)}\text{d}x\text{,}$ where $r,s,\dots$ are rational numbers, can be reduced to integrals of rational functions by means of the substitution

$\frac{\text{α}x+\text{β}}{\text{γ}x+\text{δ}}=t^m,$

FI II 57

where m is the common denominator of the fractions $r\text{,}s\text{,…}$

2.202 Integrals of the form ${\int }^{\text{}}{x}^m{(a+b{x}^n)}^{p}dx$,^∗ are rational ...