Integrate $\int\sqrt{\left(x+\frac{1}{2}\right)^2+\frac{1}{4}}dx$

 Function Plot

Plotting: $\left(x+\frac{1}{2}\right)\sqrt{\left(x+\frac{1}{2}\right)^2+\frac{1}{4}}+\frac{1}{8}\ln\left(2\sqrt{\left(x+\frac{1}{2}\right)^2+\frac{1}{4}}+2\left(x+\frac{1}{2}\right)\right)+\frac{\left(-\frac{1}{2}-x^2-x\right)\left(x+\frac{1}{2}\right)}{2\sqrt{\left(x+\frac{1}{2}\right)^2+\frac{1}{4}}}+C_0$