Integrals of the Form $\int\dfrac{dx}{ax^2+bx+c}$ and $\int\dfrac{dx}{\sqrt{ax^2+bx+c}}$

To evaluate $\int\dfrac{dx}{ax^2+bx+c}$ or $\int\dfrac{dx}{\sqrt{ax^2+bx+c}}$, follow five steps: (1) write $ax^2+bx+c$ as $a\left(x^2+\frac bax+\frac ca\right)$ and pull $a$ (or $\sqrt a$) outside the integral; (2) complete the square on $x^2+\frac bax$ by adding and subtracting $\left(\frac12\cdot\text{coefficient of }x\right)^2$; (3) write the result as a sum or difference of two squares, $(x+\beta)^2\pm\alpha^2$ or $\alpha^2-(x+\beta)^2$; (4) use $\int f(x)\,dx=g(x)+c\ \Rightarrow\ \int f(x+\beta)\,dx=g(x+\beta)+c$; (5) apply whichever of the seven special formulae of 3.2.3 matches the resulting shape, then substitute back in terms of $x$.

$\int\dfrac{dx}{4x^2+11}$. Factor out $4$: $\int\dfrac{dx}{4\left(x^2+\frac{11}4\right)}=\frac14\int\dfrac{dx}{x^2+\left(\frac{\sqrt{11}}2\right)^2}$, a Formula-1 shape with $a=\frac{\sqrt{11}}2$: result $=\dfrac{1}{2\sqrt{11}}\tan^{-1}\dfrac{2x}{\sqrt{11}}+c$.

$\int\dfrac{dx}{\sqrt{3x^2-7}}$. Factor out $3$: $\dfrac{1}{\sqrt3}\int\dfrac{dx}{\sqrt{x^2-\left(\sqrt{7/3}\right)^2}}$, a Formula-5 shape: result $=\dfrac{1}{\sqrt3}\log\left|x+\sqrt{x^2-\frac73}\right|+c$.

$\int\dfrac{dx}{a^2-b^2x^2}$. Factor out $b^2$: $\dfrac{1}{b^2}\int\dfrac{dx}{\left(\frac ab\right)^2-x^2}$, a Formula-3 shape: result $=\dfrac{1}{2ab}\log\left|\dfrac{a+bx}{a-bx}\right|+c$.

$\int\dfrac{dx}{x^2+8x+12}$. Complete the square: $x^2+8x+16-4=(x+4)^2-2^2$, a Formula-2 shape: result $=\dfrac14\log\left|\dfrac{x+2}{x+6}\right|+c$.

$\int\dfrac{dx}{\sqrt{3x^2-4x+2}}$. Factor out 3: $\frac{1}{\sqrt3}\int\dfrac{dx}{\sqrt{x^2-\frac43x+\frac23}}$. Complete the square: $x^2-\frac43x+\frac49-\frac49+\frac23=\left(x-\frac23\right)^2+\frac29$, a Formula-6 shape: result $=\dfrac{1}{\sqrt3}\log\left|x-\frac23+\sqrt{x^2-\frac43x+\frac23}\right|+c$. …