An integration reunion

Since the integrand of $\displaystyle \int_1^{\sqrt 3} \frac {5000}{x^2(x^2-4x+5)^3} dx = \frac {P(x)}{Q(x)}$ is of the form of perfect fraction where denominator has repeated roots, Ostrogradsky's method applies. Then the solution is of the form:

$\int \frac {P(x)}{Q(x)} dx = \frac {P_1(x)}{Q_1(x)} + \int \frac {P_2(x)}{Q_2(x)} dx$

where $Q(x) =x^2(x^2-4x+5)^3$ , $Q'(x) = 2x(x^2-4x+5)^3 + 3x^2(2x-4)(x^2-4x+5)^2$ , the greatest common divisor of $Q(x)$ and $Q'(x)$ is $Q_1(x) = x(x^2-4x+5)^2$ and $Q_2(x) = \dfrac {Q(x)}{Q_1(x)} = x(x^2-4x+5)$ . Let $P_1(x) = Ax^4+Bx^3+Cx^2+Dx+E$ , one degree less than $Q_1(x)$ , and $P_2(x) = Fx^2 + Gx + H$ , one degree less than $Q_2(x)$ . Then we have:

$\int \frac {5000}{x^2(x^2-4x+5)^3} dx = \frac {Ax^4+Bx^3+Cx^2+Dx+E}{x(x^2-4x+5)^2} + \int \frac {Fx^2+Gx+H}{x(x^2-4x+5)} dx$

Differentiate both sides with respect to $x$ and rearrange.

$\begin{aligned} \frac {5000}{x^2(x^2-4x+5)^3} & = \frac {(4Ax^3+3Bx^2+2Cx+D)x(x^2-4x+5)^2-(Ax^4+Bx^3+Cx^2+Dx+E)(5x^4-32x^3+78x^2-80x+25)}{x^2(x^2-4x+5)^4} + \frac {Fx^2+Gx+H}{x(x^2-4x+5)} \\ 5000(x^2-4x+5) & = (4Ax^3+3Bx^2+2Cx+D)x(x^2-4x+5)^2-(Ax^4+Bx^3+Cx^2+Dx+E)(5x^4-32x^3+78x^2-80x+25)+ (Fx^2+Gx+H)x(x^2-4x+5)^3 \end{aligned}$

By equating coefficients (see note below): We have

$\begin{aligned} \int_0^{\sqrt 3} \frac {5000}{x^2(x^2-4x+5)^3} dx & = \frac {405x^4-2190x^3+4255x^2-2150x-1000}{x(x^2-4x+5)^2} \bigg|_0^{\sqrt 3} + \int_0^{\sqrt 3} \frac {405x+480}{x(x^2-4x+5)} dx \end{aligned}$

$$

• Equating the coefficients of $x^0$ $\implies - 25 E = 25000 \implies E = -1000$
• Equating the coefficients of $x^9$ $\implies F = 0$
• Equating the coefficients of $x^8$ $\implies -A+G = 0 \implies A=G$