What is $\int \frac{x}{x^5-7} dx$? I have tried out many trigonometric substitution like $x=\sin^{\frac{2}{5}}z$. But it did not work.

To make the problem more general, consider
$$I_n=\int\frac x {x^n-A}\,dx=A^{\frac 2n-1}\int\frac t {t^n-1}\,dt=A^{\frac 2n-1}\int\frac t {\prod_{i=1}^n(t-r_i)}\,dt$$ where $r_i$ are the roots of unity. Using partial fraction decomposition, you will end with
$$I_n=A^{\frac 2n-1}\sum _{i=1}^n\int \frac {\alpha_i}{t-r_i}\,dt=A^{\frac 2n-1}\sum _{i=1}^n{\alpha_i}\log({t-r_i})$$ For sure, since the ${\alpha_i},r_i$ terms will be complex numbers, you will need to recombine them by pairs if you want to get rid of all complex terms (coefficients and logarithms).
There is also a shorter notation
$$J_n=\int\frac t {t^n-1}\,dt=-\frac{1}{2} t^2 \, _2F_1\left(1,\frac{2}{n};1+\frac{2}{n};t^n\right)$$ where appears the Gaussian or ordinary hypergeometric function.
