How to compute $\int \frac{16 x^3 - 42 x^2+2x}{\sqrt{-16x^8+112x^7-204x^6+28x^5-x^4+1}}\,\mathrm dx.$

Question

I want to compute the following integral:

$$\int \frac{16 x^3 - 42 x^2+2x}{\sqrt{-16x^8+112x^7-204x^6+28x^5-x^4+1}}\,\mathrm dx.$$

First I tried substituting $y=\text{denominator}$ but it gets very messy.

Also, I tried using partial fractions but it doesn’t work because of the square root.

What can I do here?

Answer 1

HINT

Note the expression under the square root in the denominator can be rewritten as $1-(4x^4-14x^3+x^2)^2$ and now the numerator is exactly the derivative of the polynomial inside the parentheses!

Answer 2

$-16x^8+112x^7-204x^6+28x^5-x^4+1=-(4x^4-14x^3+x^2+1)(4x^4-14x^3+x^2-1)$

Set $a=4x^4-14x^3+x^2.$ The integral rewrites $$\int \frac{\mathrm da}{\sqrt{-a^2+1}}.$$