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**Spacemoss** Do you have to try it by Integration of parts? If not, the following integrals from a list of integrals may be easier. I think those are the form you need.

Assume (ax2 + bx + c) cannot be reduced to the following expression (px + q)2 for some p and q.

$\displaystyle R = \sqrt{ax^2+bx+c}$

$\displaystyle \int R\,dx= \frac{2ax+b}{4a} R+ \frac{4ac-b^{2}}{8a} \int \frac{dx}{ R}$

$\displaystyle \int\frac{dx}{R} = \frac{1}{\sqrt{a}}\ln\left|2\sqrt{a}R+2ax+b\right| \qquad \mbox{(for }a>0\mbox{)}$

First simplify $\displaystyle \sqrt{1+(4/9)x^2},$ which should be $\displaystyle (1/3)\sqrt{4x^2+9},$, pull out the 1/3 and save it until the end, then use the formulas with a = 4, b = 0, and c = 9.