\int \frac{x+\sqrt{2}}{x^2+\sqrt{2} x+1} \, dx=\frac{1}{2}\int \frac{2x+2\sqrt{2}}{x^2+\sqrt{2} x+1} \, dx=\frac{1}{2}\int \frac{2x+\sqrt{2}}{x^2+\sqrt{2} x+1} \, dx+\frac{1}{2}\int \frac{\sqrt{2}}{x^2+\sqrt{2} x+1} \, dx