The integral ∫3x¹³+2x¹¹/(2x⁴+3x²+1)⁴ dx is equal to: (where C is a constant of integration)

∫3x¹³+2x¹¹/(2x⁴+3x²+1)⁴dx
∫3x³+2x⁵/(2x⁴+3x²+1)⁴dx
Let (2x⁴+3x²+1) = t
Then 8x³+6x dx = dt
=> 2(4x³+3x)dx = dt
=> (4x³+3x)dx = dt/2
Also, 3x³+2x⁵ = x(3x²+2x⁴) = x(t-1)/2
Then ∫ x(t-1)/2t⁴ (dt/2) = 1/4 ∫x(t-1)/t⁴ dt
This method is not correct.
Let's try another approach.
Let u = 2x⁴+3x²+1
du = (8x³+6x)dx = 2x(4x²+3)dx
∫3x¹³+2x¹¹/(2x⁴+3x²+1)⁴dx = ∫x¹¹(3x²+2)/(2x⁴+3x²+1)⁴ dx
Let's substitute u = 2x⁴+3x²+1
Then du = (8x³+6x)dx
We can rewrite the integral as:
∫x¹¹ (3x²+2) / u⁴ dx
We have a problem because we cannot easily express x¹¹ in terms of u.
Let's try a different substitution.
Let u = 2x⁴ + 3x² + 1
Then du/dx = 8x³ + 6x = 2x(4x² + 3)
The integral becomes difficult to solve using this substitution.
Let's use a different approach.
Let's try to rewrite the numerator in terms of the derivative of the denominator:
∫(3x¹³+2x¹¹)/(2x⁴+3x²+1)⁴ dx
Let t = 2x⁴ + 3x² + 1
Then dt = (8x³ + 6x)dx = 2x(4x² + 3)dx
We can see that the numerator is not easily expressible in terms of dt. This integral appears to be quite complex and may not have a simple closed-form solution using elementary functions. More advanced techniques like integration by parts or partial fraction decomposition might be required, depending on the form of the integrand. The provided solution is incorrect. There is no simple substitution that will solve this integral.