Find the integral $\int\frac{\frac{1}{2}x\left(1+x\right)}{x}dx$

Find the integral

$\int\frac{dx}{\left(5-4x-x^2\right)^{\frac{3}{2}}}$

Find the integral

$\int\frac{x}{\left(4x^2+3\right)^6}dx$

Find the integral

$\int\frac{x^2}{\sqrt{4+x^2}}dx$

Find the integral

$\int\left(dx\div x^2\sqrt{9-x^2}\right)$

Find the integral

$\int\frac{x^2dx}{\left(9-x^2\right)^{\frac{3}{2}}}$

Find the integral

$\int\frac{dx}{x\sqrt{x^{2}+36}}$

Find the integral

$\int\frac{\left(x^2-16\right)^{\frac{3}{2}}}{x^3}dx$