# Use substitution to convert the integrals to integrals of rational functions. Then use partial fractions to evaluate the integrals. int (dx)/(sqrt(x)+root(4)(x))

Use substitution to convert the integrals to integrals of rational functions. Then use partial fractions to evaluate the integrals. $\int \frac{dx}{\sqrt{x}+\sqrt[4]{x}}$
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$\int \frac{dx}{\sqrt{x}+\sqrt[4]{x}}$
Let $x={u}^{4}⇒dx=4{u}^{3}du$
Apply the substitution
$\int \frac{dx}{\sqrt{x}+\sqrt[4]{x}}=\int \frac{4{u}^{3}}{{u}^{2}+u}du$
Simplify
$\int \frac{4{u}^{2}}{u+1}du$
By the long division
$\int \frac{4{u}^{2}}{u+1}du=\int \left(4u-4+\frac{4}{u+1}\right)du$
Integrate
$2{u}^{2}-4u+4\mathrm{ln}|u+1|+C$
Back - substitute $x={u}^{4}⇒u=\sqrt[4]{x}$
$2\left(\sqrt[4]{x}{\right)}^{2}-4\sqrt[4]{x}+4\mathrm{ln}|\sqrt[4]{x}+1|+C$
$2{x}^{1/2}-4\sqrt[4]{x}+4\mathrm{ln}|\sqrt[4]{x}+1|+C$
Result:
$2{x}^{1/2}-4\sqrt[4]{x}+4\mathrm{ln}|\sqrt[4]{x}+1|+C$