The integral of the form$\int x^m{(a+b{x}^n)}^{p}dx,$ where m, n, p are rational numbers, is expressed through elementary functions only in the following cases:

     I: p is a positive integer 

In this case$(a+bxn)p$ can be expanded by the binomial theorem and the integrand can be expressed as a sum of rational powers of xn in the form $a_0=a_{1}xn+\mathrm{...}+apxnp,$which can be integrated easily.

II: p is a negative integer 

Put x = zk, where k is the common denominator of the fractions m and n

III: $\frac{m+1}{n}$is an integer 

Put $a+bxn=zk,$where k is the denominator of the fraction p.

IV: $\frac{m+1}{n}+p$is an integer 

Put $a+bxn=xnzk,$where k is the denominator of the fraction p.