Evaluate: `intsqrt(2x^2+3x+4)\ dx`

Explanation

`int frac{3 x+1}{2 x^{2}-2 x+3} d x=frac{1}{2} int frac{3 x+1}{x^{2}-x+frac{3}{2}} d x=frac{3}{2} int frac{x+frac{1}{3}}{x^{2}-x+frac{3}{2}} d x^{2}`

`=frac{3}{4} int frac{2 x+frac{2}{3}}{x^{2}-x+frac{3}{2}} d x=frac{3}{4} int frac{2 x-1+1+frac{2}{3}}{x^{2}-x+frac{3}{2}} d x`

`=frac{3}{4} int frac{2 x-1+frac{5}{3}}{x^{2}-x+frac{3}{2}} d x=frac{3}{4} int frac{2 x-1}{x^{2}-x+frac{3}{2}} d x+frac{3}{4} times frac{5}{3} int frac{d x}{x^{2}-x+frac{3}{2}} `

`(`because` int[c_{1} f(x)+c_{2} g(x) d x]=c_{1} int f(x) d x+c_{2} int g(x) d x)`

`=frac{3}{4} int frac{2 x-1}{x^{2}-x+frac{3}{2}} d x+frac{5}{4} int frac{d x}{(x-frac{1}{2})^{2}+frac{3}{2}-frac{1}{4}} `

`quad(`because`(a-b)^{2}=a^{2}+b^{2}-2 a b) `

`=frac{3}{4} int frac{2 x-1}{x^{2}-x+frac{3}{2}} d x+frac{5}{4} int frac{d x}{(x-frac{1}{2})^{2}+frac{5}{4}}=frac{3}{4} int frac{2 x-1}{x^{2}-x+frac{3}{2}} d x+frac{5}{4} int frac{d x}{(x-frac{1}{2})^{2}+(frac{sqrt{5}}{2})^{2}} `

`=frac{3}{4} log |x^{2}-x+frac{3}{2}|+frac{5}{4} times frac{1}{frac{sqrt{5}}{2}} tan ^{-1}(frac{x-frac{1}{2}}{frac{sqrt{5}}{2}})+C^{prime},`

...