integration of fraction power expressions

The antiderivatives of every expression containing fraction powers can not be expressed by using elementary functions. However, there are after making a substitution.

• ${\displaystyle \int R(x,x^{r_1},\mathrm{\dots },x^{r_m})𝑑x}$,  where $R$ means a rational function of its arguments.If the common denominator of the fraction power exponents $r_j$ is $n$, the substitution

  $$x:=t^n,dx=n{t}^{n-1}dt$$

  changes each exponent to an integer and the whole integrand to a rational function in the variable $t$.
  

  Example.  For  ${\displaystyle \int \frac{x^{\frac{1}{2}}}{x^{\frac{3}{4}}+1}𝑑x}$  the least common multiple of the denominators of $\frac{1}{2}$ and $\frac{3}{4}$ is 4, whence we make the substitution  $x=t^4$,  $dx=4t^{3}dt$.  Then we obtain

  $$\int \frac{x^{\frac{1}{2}}}{x^{\frac{3}{4}}+1}𝑑x=\mathrm{\hspace{0.33em}4}\int \frac{t^{5}dt}{t^3+1}=\mathrm{\hspace{0.33em}4}\int \left(t^2-\frac{t^2}{t^3+1}\right)𝑑t=\mathrm{\hspace{0.33em}4}\left(\frac{t^3}{3}-\frac{1}{3}\ln |t^3+1|\right)+C$$

  $$=\frac{4}{3}\left(x^{\frac{3}{4}}-\ln |x^{\frac{3}{4}}+1|\right)+C.$$

• In ${\displaystyle \int R(x,{\left(\frac{ax+b}{cx+d}\right)}^{r_1},\mathrm{\dots },{\left(\frac{ax+b}{cx+d}\right)}^{r_m})𝑑x}$,  correspondently the substitution

  $$\frac{ax+b}{cx+d}:=t^n$$

  changes the integrand to a rational function.
  

  Example.  For  ${\displaystyle \int \frac{\sqrt{x+4}}{x}𝑑x}$  we substitute  $x+4=t^2$,  $dx=2tdt$,  getting

  $$\int \frac{\sqrt{x+4}}{x}𝑑x=\mathrm{\hspace{0.33em}2}\int \frac{t^2}{t^2-4}𝑑t=\mathrm{\hspace{0.33em}2}\int \left(1+\frac{4}{t^2-4}\right)𝑑t=\mathrm{\hspace{0.33em}2}t+2\ln \left|\frac{t-2}{t+2}\right|+C$$

  $$=\mathrm{\hspace{0.33em}2}\sqrt{x+4}+2\ln \left|\frac{\sqrt{x+4}-2}{\sqrt{x+4}+2}\right|+C.$$