Simpliest integrals with quadratic trinomials are usually being reduced to well known forms.
$$\int\frac{Ax+B}{ax^2+bx+c}dx\longrightarrow \int\frac{A'u+B'}{au^2+m}du\,\,,\,\small{u=x+\frac{b}{2a}}$$
$$\int\frac{Ax+B}{\sqrt{ax^2+bx+c}}dx\longrightarrow \int\frac{A'u+B'}{\sqrt{au^2+m}}du\,\,,\,\small{u=x+\frac{b}{2a}}$$
$$\int (Ax+B)\sqrt{ax^2+bx+c}\,\,dx\longrightarrow \int Au\sqrt{au^2+m}\,\,du+\int B'\sqrt{au^2+m}\,\,du\,\,,\,\small{u=x+\frac{b}{2a}}$$
$$\int\frac{dx}{(x-\vartheta)\sqrt{ax^2+bx+c}} \longrightarrow -\int\frac{d\delta}{\sqrt{a'\delta^2+b'\delta+c'}} \,\,,\,\small{\delta=\frac{1}{x-\vartheta}}$$
$$\int\sqrt{a^2-x^2}dx=\frac{x\sqrt{a^2-x^2}}{2}+\frac{a^2}{2}\arcsin{\frac{x}{a}}$$
$$\int\sqrt{x^2\pm a^2}dx=\frac{x\sqrt{x^2\pm a^2}}{2}+\frac{a^2}{2}\ln{\big|x+\sqrt{x^2\pm a^2}\big|}$$
1
$$\int\frac{2x-5}{x^2+x+3}dx$$
2
$$\int\frac{7-3x}{\sqrt{x^2+x+1}}dx$$
3
$$\int\frac{4x-11}{\sqrt{1+x-x^2}}dx$$
4
$$\int\frac{\ln{x}+2}{x\sqrt{1-\ln{x}-\ln^2{x}}}dx$$
5
$$\int\frac{e^{2x}+3e^x}{\sqrt{e^{2x}+e^x+1}}dx$$
6
$$\int\frac{e^{2x}}{e^{2x}+2e^x+4}dx$$
7
$$\int\frac{dx}{\sqrt{e^{2x}-5e^x+6}}$$
8
$$\int\frac{dx}{(x+1)\sqrt{x^2+1}}$$
9
$$\int\frac{dx}{(x-1)\sqrt{4x^2-10x+7}}$$
10
$$\int (x+2)\sqrt{x^2+x+1}\,\,dx$$
11
$$\int(1-3x)\sqrt{1+x-x^2}\,dx$$
12
$$\int\frac{x^3+x}{-1-x^2+x^4}dx$$
13
$$\int\frac{x-x^3}{\sqrt{1+x^2+x^4}}dx$$
14
$$\int\frac{x^3dx}{x^4-x^2+5}$$
15
$$\int\frac{dx}{x\sqrt{x^4+x^2+1}}$$
16
$$\int\frac{(x+1)^2}{x\sqrt{x^2+3x+1}}dx$$
17
$$\int(x^3+x)\sqrt{1+x^4}\,dx$$
18
$$\int\cos{x}\sqrt{\cos{2x}}\,dx$$
19
$$\int\sin{x}\sqrt{\cos{2x}}\,dx$$
20
$$\int\frac{1+\sin{x}}{\cos{x}\sqrt{\cos{2x}}}dx$$
