$\int \dfrac{dx}{x^{2}+4x+5}$

$\int \dfrac{dx}{x^{2}+2x+5}$

$\int \dfrac{dx}{x^{2}+4x+8}$

$\int \dfrac{dx}{x^{2}-6x+10}$

$\int \dfrac{2\,dx}{3+2x+2x^{2}}$

$\int \dfrac{3\,dx}{x^{2}+6x+10}$

$\int \dfrac{x\,dx}{2x^{2}+2x+3}$

$\int \dfrac{3x\,dx}{x^{2}+6x+10}$

$\int \dfrac{dx}{\sqrt{8+2x-x^{2}}}$

$\int \dfrac{dx}{\sqrt{5-4x-2x^{2}}}$

$\int \dfrac{dx}{\sqrt{4x-x^{2}}}$

$\int \dfrac{dx}{\sqrt{x^{2}-6x-10}}$

$\int \dfrac{dx}{( x+1) \sqrt{x^{2}+2x+2}}$

$\int \dfrac{dx}{( x-4) \sqrt{x^{2}-8x+17}}$

$\int \dfrac{dx}{\sqrt{24-2x-x^{2}}}$

$\int \dfrac{dx}{\sqrt{9x^{2}+6x+10}}$

$\int \dfrac{x-5}{\sqrt{x^{2}-2x+5}}dx$

$\int \dfrac{x+1}{x^{2}-4x+3}dx$

$\int_{1}^{3}\dfrac{dx}{\sqrt{x^{2}-2x+5}}$

$\int_{1/2}^{1}\dfrac{x^{2}\,dx}{\sqrt{2x-x^{2}}}$

$\int \dfrac{e^{x}\,dx}{\sqrt{e^{2x}+e^{x}+1}}$

$\int \dfrac{\cos x\,dx}{\sqrt{\sin ^{2}x+4\sin x+3}}$

$\int \dfrac{2x-3}{\sqrt{4x-x^{2}-3}} dx$

$\int \dfrac{x+3}{\sqrt{x^{2}+2x+2}}\,dx$

$\int \dfrac{dx}{(x^{2}-2x+10)^{3/2}}$

$\int \dfrac{dx}{\sqrt{x^{2}-2x+10}}$

$\int \dfrac{dx}{\sqrt{x^{2}+2x-3}}$

$\int x\sqrt{x^{2}-4x-1} dx$

$\int \dfrac{\sqrt{5+4x-x^{2}}}{x-2}\,dx$

$\int \sqrt{5+4x-x^{2}}\,dx$

$\int \dfrac{x~dx}{\sqrt{x^{2}+2x-3}}$

$\int \dfrac{x~dx}{\sqrt{x^{2}-4x+3}}$

Show that if $k\gt0$, then $$\int \dfrac{dx}{\sqrt{(x+h)^{2}+k} }=\ln \left[ \sqrt{(x+h)^{2}+k}+x+h\right] +C$$

Show that if $a\gt0$ and $b^{2}-4ac\gt0$, then $$\int \dfrac{dx}{\sqrt{ax^{2}+bx+c}}=\dfrac{1}{\sqrt{a}}\ln \left\vert \sqrt{ax^{2}+bx+c}+\sqrt{a}x+\dfrac{b}{2\sqrt{a}}\right\vert +\,C$$

Find $\int \sqrt{\dfrac{a+x}{a-x}}\, dx$, where $a > 0$.