The integral problem: $\int_{0}^{16}\frac{dx}{\sqrt{x^2+9}-\sqrt{x}}$ I have met this kind of problem today.in fact, I spent hours trying to solve this problem on its own. My method was just to calculate the indefinite integral. I looked at Wolfram Alpha after I failed. Wolfram couldn't evaluate this integral. I don't know the spesific reason.

The integral:
$$\int_{0}^{16}\frac{dx}{\sqrt{x^2+9}-\sqrt{x}}$$



According to Maple 18, the integral is equal to
\begin{align*}
&\frac15\arctan \left( \frac{\sqrt {5}}{5} \right) \sqrt {5}+\frac{1}{14}\sqrt {7} \ln  \left( \sqrt {7}+1 \right) -\frac{1}{14}\sqrt {7}\ln  \left( \sqrt {7}-1 \right) \\
&-\frac{1}{14}\sqrt {7}\ln  \left( \sqrt {5}\sqrt {7}\sqrt {53}+19 \right) +\frac{1}{14}\sqrt {7}\ln  \left( \sqrt {5}\sqrt {7}\sqrt {53}-19 \right) \\
&+\frac{1}{5}\sqrt {5}\arctan \left( {\frac {13\,\sqrt {53}}{265}} \right) 
+\frac{1}{5}\sqrt {5}\arctan \left( \frac85\sqrt {5}-\frac{1}{5}\sqrt {7}\sqrt {5} \right) \\
&+\frac{1}{5}\sqrt {5}\arctan \left( \frac85\sqrt {5}+\frac{1}{5} \sqrt {7}\sqrt {5} \right)+\frac{1}{14}\sqrt {7}\ln  \left( 19-4\,\sqrt {7} \right) \\
&-\frac{1}{14}\sqrt {7}\ln  \left( 19+4\,\sqrt {7} \right) +\ln \left( 3 \right) -\ln  \left( -16+\sqrt {5}\sqrt {53} \right).
\end{align*}
Enjoy.
