Evaluate the following integrals:

Make perfect square of quadratic equation

$4 x^{2}+5 x+6=4\left[\left(x+\frac{5}{8}\right)^{2}+\frac{71}{64}\right]$

$y=2 \int(2 x+3) \sqrt{\left[\left(x+\frac{5}{8}\right)^{2}+\left(\frac{\sqrt{71}}{8}\right)^{2}\right]} d x$

Let, $x+\frac{5}{8}=t \Rightarrow x=t-\frac{5}{8}$

Differentiate both side with respect to $t$

$\frac{d x}{d t}=1 \Rightarrow d x=d t$

$y=2 \int\left(2 t+\frac{7}{4}\right) \sqrt{\left[t^{2}+\left(\frac{\sqrt{71}}{8}\right)^{2}\right]} d t$

$A=4 \int t \sqrt{\left(\frac{\sqrt{71}}{8}\right)^{2}+t^{2}} d t$

Let, $t^{2}=z$

Differentiate both side with respect to $z$

$2 t \frac{d t}{d z}=1 \Rightarrow t d t=\frac{1}{2} d z$

$A=2 \int \sqrt{\left(\frac{\sqrt{71}}{8}\right)^{2}+z} d z$

$A=\frac{1}{4} \int \sqrt{71+64 z} d z$

$A=\frac{1}{384}(71+64 z)^{\frac{3}{2}}+c_{1}$

Put $z=t^{2}$ and $t=x+\frac{5}{8}$

$A=\frac{1}{384}\left(71+64\left(x+\frac{5}{8}\right)^{2}\right)^{\frac{3}{2}}+c_{1}$

$A=\frac{1}{6}\left(4 x^{2}+5 x+6\right)^{\frac{3}{2}}+c_{1}$

$B=\int \frac{7}{2} \sqrt{\left(\frac{\sqrt{71}}{8}\right)^{2}+t^{2}} d t$

$B=\frac{7}{2}\left(\frac{t}{2} \sqrt{\left(\frac{\sqrt{71}}{8}\right)^{2}+t^{2}+\frac{\left(\frac{\sqrt{71}}{8}\right)^{2}}{2}} \ln \left(t+\sqrt{\left(\frac{\sqrt{71}}{8}\right)^{2}+t^{2}}\right)\right)+c_{2}$

Put $t=x+\frac{5}{8}$

$B=\frac{7}{2}\left(\frac{\left(x+\frac{5}{8}\right)}{2} \sqrt{\left(\frac{\sqrt{71}}{8}\right)^{2}+\left(x+\frac{5}{8}\right)^{2}}\right)+$

$\frac{7\left(\frac{\sqrt{71}}{8}\right)^{2}}{4} \ln \left(\left(x+\frac{5}{8}\right)+\sqrt{\left(\frac{\sqrt{71}}{8}\right)^{2}+\left(x+\frac{5}{8}\right)^{2}}\right)+c_{2}$

$B=\frac{7}{2}\left(\frac{(8 x+5)}{32} \sqrt{4 x^{2}+5 x+6}\right)+$

$\frac{497}{256} \ln \left(\left(x+\frac{5}{8}\right)+\sqrt{\left(\frac{\sqrt{71}}{8}\right)^{2}+\left(x+\frac{5}{8}\right)^{2}}\right)+c_{2}$

The final answer is $y=A+B$'

$y=\frac{1}{6}\left(4 x^{2}+5 x+6\right)^{\frac{3}{2}}+\frac{7}{2}\left(\frac{(8 x+5)}{32} \sqrt{4 x^{2}+5 x+6}\right)+$

$\frac{497}{256} \ln \left(\left(x+\frac{5}{8}\right)+\sqrt{x^{2}+\frac{5}{4} x+\frac{3}{2}}\right)+c$

$y=\frac{1}{192}\left(128 x^{2}+328 x+297\right) \sqrt{4 x^{2}+5 x+6}+$

$\frac{497}{256} \ln \left(\left(x+\frac{5}{8}\right)+\sqrt{x^{2}+\frac{5}{4} x+\frac{3}{2}}\right)+c$