Integration of fraction with square root I have a problem with integrating of fraction
$$
\int \frac{x}{x^2 + 7 + \sqrt{x^2 + 7}}
$$
I have tried to rewrite it as $\int \frac{x^3 + 7x - x \sqrt{x^2 + 7}}{x^4 + 13x^2 + 42} = \int \frac{x^3 + 7x - x \sqrt{x^2 + 7}}{(x^2 + 6)(x^2 + 7)}$ and then find some partial fractions, but it wasn't succesful.

Consider the integral
\begin{align}
I = \int \frac{x}{x^{2} + 7 + \sqrt{x^2 + 7}} \, dx
\end{align}
Make the substitution $u = \sqrt{x^{2} + 7}$ for which $x = \sqrt{u^{2} - 7}$ and the integral becomes
\begin{align}
I &= \int \frac{\sqrt{u^{2}-7}}{u^{2} + u} \cdot \frac{u \, du}{\sqrt{u^{2}-7}} \\
&= \int \frac{du}{1+u} \\
&= \ln(1+u)
\end{align}
Making the reverse substitution yields
\begin{align}
\int \frac{x}{x^{2} + 7 + \sqrt{x^2 + 7}} \, dx = \ln(1 + \sqrt{x^{2} + 7}).
\end{align}
