Answer
$\dfrac{53}{6}$
Work Step by Step
The formula to calculate the arc length is as follows:
$L=\int_m^n \sqrt {1+[f'(x)]^2} dx$
Re-write the equation as follows:
$[f'(x)]^2=\dfrac{(y-1)^2}{4y}$
This implies that
$L=\int_{0}^{2} [\dfrac{(2x^2+2x+)^2(2x^2+6x+5)^2}{16(x+1)^4}] dx $
Separate the terms and integrate as follows:
$L=\int_{0}^{2} x^2+2x+1+\dfrac{1}{4(x+1)^2} dx \\=[\dfrac{x^{3}}{3}+x^2+x-\dfrac{1}{4(x+1)}]_{0}^{2}=\dfrac{53}{6}$
