\(\int \frac{x^2}{x^2 + 3x - 3}dx \)
\(= \int \frac{x^2+3x -3-(3x-3)}{x^2 + 3x - 3}dx\)
\(= \int \left(1 - \frac{3x - 3}{x^2 + 3x - 3}\right)dx\)
\(= x - \frac 32 \int \frac{2x+3-5}{x^2+3x-3}dx\)
\(= x - \frac 32 \int \frac{2x+3}{x^2+3x-3}dx + \frac {15}2 \int \frac{dx}{(x + \frac 32)^2 - \frac{21}4}+ C\)
\(= x - \frac 32 \log(x^2 + 3x - 3) + \frac{15}2 \times \frac 1{2\frac{\sqrt {21}}2} \log \left( \cfrac{x + \frac 32 - \frac{\sqrt{21}}2}{x + \frac 32 +\frac{\sqrt{21}}2}\right)+C\)
\(= x - \frac 32 \log (x^2 +3x - 3) + \frac{5\sqrt3}{2\sqrt 7}\log \left|\frac{2x+3-\sqrt{21}}{2x +3+\sqrt{21}}\right|+C\)
