How Do You Manually Integrate \(\frac{dx}{x(a\sqrt{x}+b)}\)?

[Grogs]

I ran into the following (general form) integral today while working on an engineering problem:

$$\int \frac{dx}{x (a \sqrt{x} +b)}$$

I ran it through 'integrals.com' and came up with the following solution:

$$\frac {\ln{x} - 2 \ln{(a \sqrt{x} + b)}}{b}$$

which is fine for the purpose of solving the problem, but I'd like to be able to work it by hand. I tried integration by parts, but any choice of u and dv I could think of only made it messier. I also couldn't think of an appropriate u substitution.

Thanks in advance,

Grogs

 

[arildno]

Try $u=\sqrt{x}$
With the use of partial fractions decomposition, the result easily follows.

 

[Grogs]

Thanks arildno.

Partial fractions is one of those skills I relearn once a year or so when I need to do a LaPlace Transform and then quickly forget.