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

• $\displaystyle\int R(x,\,x^{r_1},\,\ldots,\,x^{r_m})\,dx$ , where $R$ means a rational function of its arguments. If the common denominator of the fraction power exponents $r_j$ is $n$ , the substitution $$x = t^n, \quad dx = nt^{n-1}dt$$ 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 $x = t^4$ , $dx = 4t^3dt$ . Then we obtain $$\int\frac{x^{\frac{1}{2}}}{x^{\frac{3}{4}}+1}\,dx = 4\!\int\frac{t^5dt}{t^3+1} = 4\!\int\left(t^2-\frac{t^2}{t^3+1}\right)dt = 4\left(\frac{t^3}{3}-\frac{1}{3}\ln|t^3+1|\right)+C$$ $$= \frac{4}{3}\left(x^{\frac{3}{4}}-\ln|x^{\frac{3}{4}}+1|\right)+C.\\$$

• In $\displaystyle\int R\left(x,\,\left(\frac{ax+b}{cx+d}\right)^{r_1},\,\ldots,\,\left(\frac{ax+b}{cx+d}\right)^{r_m}\right)\,dx$ , 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 $x+4 = t^2$ , $dx = 2t\,dt$ , getting $$\int\frac{\sqrt{x+4}}{x}\,dx = 2\!\int\frac{t^2}{t^2-4}\,dt = 2\!\int\left(1+\frac{4}{t^2-4}\right)dt = 2t+2\ln\left|\frac{t-2}{t+2}\right|+C$$ $$= 2\sqrt{x+4}+2\ln\left|\frac{\sqrt{x+4}-2}{\sqrt{x+4}+2}\right|+C.$$

Bibliography

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N. PISKUNOV: Diferentsiaal- ja integraalarvutus kõrgematele tehnilistele õppeasutustele. Viies, täiendatud trükk. Kirjastus ``Valgus'', Tallinn (1965).