<s>[INST] Finding an integral using a table? Am I correct for pattern matching this integral?
I have
$$\int \frac{\sqrt{9x^2+4}}{x^2}dx$$
Does this pattern match with:
$$\int \frac{\sqrt{a^2 + x^2}}{x^2}dx = -\frac{a^2 + x^2}{x} + \ln(x + \sqrt{a^2 + x^2}) + c$$
If I factor out the 9, I get
$$= 3 \int \frac{\sqrt{x^2 + \frac{4}{9}}}{x^2}$$
with $a = \frac{2}{3}$
I get:
$$3 \left( - \frac{\sqrt{\frac{4}{9}+x^2}}{x} + \ln\left(x+\sqrt{\frac{4}{9}+x^2}\right) +c\right)$$
Is this the right track?
Wolfram winds up with a different answer though:

 [/INST] You are indeed correct. Note that in an indefinite integral, perfectly valid answers can be separated by (any) constant.
The first term in Wolfram's answer can be rewritten: 
$3\ln{(\frac32(x+\sqrt{\frac49 + x^2}))} = 3\ln{(x+\sqrt{\frac49 + x^2})} + 3\ln\frac 32$
and the second term can be rearranged to be identical to your other term.
So your answers are separated by a constant. That's fine. You're right.
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