Integral of irrational function $\int \frac{dx}{x\left(\sqrt[3]{x}+\sqrt[5]{x^2}\right)}$

Question

$$\int \frac{dx}{x\left(\sqrt[3]{x}+\sqrt[5]{x^2}\right)}$$

I have tried to solve the integral using, substitution but it doesn’t lead to the correct result.

Do you folks have any ideas about which technique I should use to solve the given integral?

Answer 1

I would first attempt to remove all the fractional powers with an appropriate substitution. This suggests the choice $x = u^{15}$, so that $dx = 15 u^{14} \, du$, hence $$\int \frac{dx}{x(x^{1/3} + x^{2/5})} = \int \frac{15 u^{14} \, du}{u^{15} (u^5 + u^6)} = 15 \int \frac{du}{u^6 (1+u)}.$$ Now a partial fraction decomposition is desired: we seek constants $A, B, C, D, E, F, G$ such that $$\frac{A}{u} + \frac{B}{u^2} + \frac{C}{u^3} + \frac{D}{u^4} + \frac{E}{u^5} + \frac{F}{u^6} + \frac{G}{1+u} = \frac{1}{u^6(1+u)},$$ which is reasonably straightforward to solve. Once this is done, the rest is trivial.

Answer 2

You should use the substitution $x=y^{15}$, thereby getting a rational function:$$15\int\frac 1{y^6+y^7}\,\mathrm dy.$$

Answer 3

Plug $x=u^{15}$ so $dx=15u^{14}$

$\sqrt[3]{x}=u^5;\;\sqrt[5]{x^2}=u^6$

Integral becomes

$$15\int \frac{du}{u^7+u^6} =15\int \frac{du}{u^6 (u+1)}=15\int\left[\frac{1}{u^6}-\frac{1}{u^5}+\frac{1}{u^4}-\frac{1}{u^3}+\frac{1}{u^2}+\frac{1}{u+1}-\frac{1}{u}\right]\,du$$

Indeed with partial fraction we have: $$\frac{a}{u}+\frac{b}{u^2}+\frac{c}{u^3}+\frac{d}{u^4}+\frac{e}{u^5}+\frac{f}{u^6}+\frac{g}{u+1}=\frac{1}{u^6 (u+1)}$$

Numerator is

$u^6 (a+g)+u^5 (a+b)+u^4 (b+c)+u^3 (c+d)+u^2 (d+e)+u (e+f)+f$

which must be identical to $1$ so $f=1,\;e=-1,\; a = -1,\;b = 1,\;c = -1,\;d = 1,\;g = 1$

So the integral gives

$$15 \left(-\frac{1}{5 u^5}+\frac{1}{4 u^4}-\frac{1}{3 u^3}+\frac{1}{2 u^2}-\frac{1}{u}-\log (u)+\log (u+1)\right)+C=$$ $$=15 \left(\frac{1}{2 \sqrt[2]{x^{15}}}+\frac{1}{4 \sqrt[4]{x^{15}}}-\frac{1}{5 \sqrt[3]{x}}-\frac{1}{3 \sqrt[5]{x}}-\frac{1}{\sqrt[15]{x}}-\log \sqrt[15]{x}+\log \left(\sqrt[15]{x}+1\right)\right)+C=$$ $$=\frac{15}{2 \sqrt[2]{x^{15}}}+\frac{15}{4 \sqrt[4]{x^{15}}}-\frac{3}{\sqrt[3]{x}}-\frac{5}{\sqrt[5]{x}}-\frac{15}{\sqrt[15]{x}}+15 \log \left(\sqrt[15]{x}+1\right)-\log x+C$$