<s>[INST] Value of $(p,q)$ in indefinite integration Finding value of $(p,q)$ in  $\displaystyle \int\frac{2x^7+3x^2}{x^{10}-2x^5+1}dx=\frac{px^3+qx^8}{x^{10}-2x^5+1}+c$
what i try
$\displaystyle \int\frac{2x^7+3x^2}{x^{10}-2x^5+1}dx$
put $x=1/t$ and $dx=-1/t^2dt$
$\displaystyle -\frac{2t^5+3x^{10}}{t^{10}-2t^5+1}dt$
How do i solve it Help me please 
 [/INST] The best way I can seem to think of is differentiating both sides of the equality that gives you the following: $$\dfrac{\mathrm d}{\mathrm dx}\int\dfrac{2x^7+3x^2}{x^{10}-2x^5+1}\mathrm dx=\dfrac{\mathrm d}{\mathrm dx}\left(\dfrac{px^3+qx^8}{x^{10}-2x^5+1}\right)$$ 

$$ \begin{equation}\begin{aligned} \dfrac{\mathrm d}{\mathrm dx}\left(\dfrac{px^3+qx^8}{x^{10}-2x^5+1}\right) &= \dfrac{(3px^{2}+8qx^7)(x^{10}-2x^5+1)-(10x^9-10x^4)(px^3+qx^8)}{(x^{10}-2x^5+1)^2} \\ &= \dfrac{-2qx^{17}-7px^{12}-6qx^{12}+4px^7+8qx^7+3px^2}{(x^{10}-2x^5+1)^2}\end{aligned}\tag1\end{equation}$$

$$\begin{equation}\begin{aligned}\dfrac{\mathrm d}{\mathrm dx}\int\dfrac{2x^7+3x^2}{x^{10}-2x^5+1}\mathrm dx&=\dfrac{2x^7+3x^2}{x^{10}-2x^5+1}\\&=\dfrac{(2x^7+3x^2)(x^{10}-2x^5+1)}{(x^{10}-2x^5+1)^2}\\&=\dfrac{2x^{17}-x^{12}-4x^7+3x^2}{(x^{10}-2x^5+1)^2}\end{aligned}\end{equation}\tag2$$

Equating $(1)$ and $(2)$ and comparing gives us the following system: $$\begin{cases}q=-1\\ 7p+6q=1 \\ p+2q=-1\\ 3p=3\end{cases}\implies \bbox [5px,border:2px solid #C0A000]{\begin{array}pp=+1 \\ q=-1\end{array}}$$
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