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Answer 1

Solution:

$\sf{\dfrac{1}{x+a}+\dfrac{1}{x+b}+\dfrac{1}{x+c}+\dfrac{ax}{x^{3}+ax^{2}}+\dfrac{bx}{x^{3}+bx^{2}}+\dfrac{cx}{x^{3}+cx^{2}}}$

$\sf{\leadsto{\dfrac{1}{x+a}+\dfrac{1}{x+b}+\dfrac{1}{x+c}+\dfrac{ax}{x(x^{2}+ax)}+\dfrac{bx}{x(x^{2}+bx)}+\dfrac{cx}{x(x^{2}+cx)}}}$

$\sf{\leadsto{\dfrac{1}{x+a}+\dfrac{1}{x+b}+\dfrac{1}{x+c}+\dfrac{a}{x^{2}+ax}+\dfrac{b}{x^{2}+bx}+\dfrac{c}{x^{2}+cx}}}$

$\sf{\leadsto{\dfrac{1}{x+a}+\dfrac{a}{x^{2}+ax}+\dfrac{1}{x+b}+\dfrac{b}{x^{2}+bx}+\dfrac{1}{x+c}+\dfrac{c}{x^{2}+cx}}}$

$\sf{Multiply \ denominator \ and \ numerator \ of \ \dfrac{1}{x+a}, }$

$\sf{\dfrac{1}{x+b}, \ \dfrac{1}{x+c} \ by \ x }$

$\sf{\leadsto{\dfrac{x}{x^{2}+ax}+\dfrac{a}{x^{2}+ax}+\dfrac{x}{x^{2}+bx}+\dfrac{b}{x^{2}+bx}+\dfrac{x}{x^{2}+cx}+\dfrac{c}{x^{2}+cx}}}$

$\sf{\leadsto{\dfrac{x+a}{x^{2}+ax}+\dfrac{x+b}{x^{2}+bx}+\dfrac{x+c}{x^{2}+cx}}}$

$\sf{\leadsto{\dfrac{x+a}{x(x+a)}+\dfrac{x+b}{x(x+b)}+\dfrac{x+c}{x(x+c)}}}$

$\sf{\leadsto{\dfrac{1}{x}+\dfrac{1}{x}+\dfrac{1}{x}}}$

$\sf{\leadsto{\dfrac{1+1+1}{x}}}$

$\sf{\leadsto{\dfrac{3}{x}}}$

$\sf\purple{\tt{\therefore{Hence, \ simplified \ form \ is \ \dfrac{3}{x}.}}}$

Answer 2

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

$\frac{1}{x +a } + \frac{1}{x +b } + \frac{1}{x +c } + \frac{ax}{ {x}^{3} +a {x}^{2} } + \frac{bx}{ {x}^{3} +b {x}^{2} } + \frac{cx}{ {x}^{3} +c {x}^{2} }$

$= \frac{1}{x +a } + \frac{1}{x +b } + \frac{1}{x +c } + \frac{ax}{ {x}^{2} ( {x} +a ) } + \frac{bx}{ {x}^{2} (x +b) } + \frac{cx}{ {x}^{2}(x +c) }$

$= \frac{1}{x +a } + \frac{1}{x +b } + \frac{1}{x +c } + \frac{a}{ {x}^{} ( {x} +a ) } + \frac{b}{ {x}^{} (x +b) } + \frac{c}{ {x}^{}(x +c) }$

$= \frac{1}{x +a } +\frac{a}{ {x}^{} ( {x} +a ) } + \frac{1}{x +b } + \frac{b}{ {x}^{} (x +b) } + \frac{1}{x +c } + \frac{c}{ {x}^{}(x +c) }$

$= \frac{x + a}{ {x}^{} ( {x} +a ) } + \frac{x + b}{ {x}^{} (x +b) } + \frac{x + c}{ {x}^{}(x +c) }$

$= \frac{1}{x} + \frac{1}{x} + \frac{1}{x}$

$= \frac{3}{x}$