The integral ∫ 3 x ^ 13 + 2 x ^ 11 (2 x ^ 4 + 3 x ^ 2 + 1) ^ 4 dx ,...

The given integral can be written as

$I = ∫ \dfrac{3 . {x}^{- 3} + 2{x}^{- 5}}{{\left(2 + 3{x}^{- 2} + {x}^{- 4}\right)}^{4}}dx$

(Dividing ${x}^{16}$ to both numerator and denominator),

Let, $2 + 3{x}^{- 2} + {x}^{- 4} = t$

$⇒ \left(- 6{x}^{- 3} - 4{x}^{- 5}\right)dx = dt$

$⇒ - 2\left(3{x}^{- 3} + 2{x}^{- 5}\right)dx = dt$

$∴ I = - \dfrac{1}{2} ∫ \dfrac{dt}{{t}^{4}} = - \dfrac{1}{2}\left(\dfrac{{t}^{- 4 + 1}}{- 4 + 1}\right) = \dfrac{1}{6}\left(\dfrac{1}{{t}^{3}}\right) + C$, where $C$ is the constant of integration.

$= \dfrac{1}{6{\left(2 + 3{x}^{- 2} + {x}^{- 4}\right)}^{3}} + C = \dfrac{{x}^{12}}{6{\left(2{x}^{4} + 3{x}^{2} + 1\right)}^{3}} + C$