How to solve the indefinite integral $\int \frac{x^{1 / 7}+x^{1 / 2}}{x^{8 / 7}+x^{1 / 14}} \ \mathrm{d} x$?

Question

I need to calculate a complicated indefinite integral below: $$\int \frac{x^{1 / 7}+x^{1 / 2}}{x^{8 / 7}+x^{1 / 14}} \ \mathrm{d} x$$ Can you help me to find out how it solved?I think my answer by using the exchange methods three times: $$ 70\ln \left|4x^{5/7}-4x^{5/14}+4\right|-\frac{1}{\sqrt{3}}\arctan \left(\frac{2x^{5/14}-1}{\sqrt{3}}\right) + x^{5/14} + C $$ is wrong.

Any help would be greatly appreciated!

Answer 1

You should apply the substitution: $x = t^{14}$

This will get you a rational function of $t$.

And then... as we know all rational functions can be integrated,
there is a well-known procedure for that.

Answer 2

Hint:

As $[2,7,14]=14,$

Let $x^{1/14}=y, x=y^{14}, dx=14y^{13}dy$

$$\int\dfrac{y^2+y^7}{y^{16}+y}\cdot14y^{13}dy$$

$$=14\int\dfrac{y^{15}(1+y^5)}{y(1+y^{15})}dy$$

$$=14\int\dfrac{y^{10}}{1-y^5+y^{10}}\cdot y^4dy$$

Set $y^5=z$

The rest is left as an exercise!