What is expand x/(x^2+1)+1/(3x)-1/(x^2) ?

The solution to expand x/(x^2+1)+1/(3x)-1/(x^2) is (4x^3+x-3x^2-3)/(3x^4+3x^2)

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expand x/(x^2+1)+1/(3x)-1/(x^2)

expand

\frac{x}{x ^{2} + 1} + \frac{1}{3 x} - \frac{1}{x ^{2}}

\frac{4 x ^{3} + x - 3 x ^{2} - 3}{3 x ^{4} + 3 x ^{2}}

Solution steps

\frac{x}{x ^{2} + 1} + \frac{1}{3 x} - \frac{1}{x ^{2}}

Least Common Multiplier of

x^{2} + 1, 3 x, x^{2} : 3 x^{2} (x^{2} + 1)

Adjust Fractions based on the LCM

= \frac{3 x ^{3}}{3 x ^{2} ( x ^{2} + 1 )} + \frac{x ( x ^{2} + 1 )}{3 x ^{2} ( x ^{2} + 1 )} - \frac{3 ( x ^{2} + 1 )}{3 x ^{2} ( x ^{2} + 1 )}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{3 x ^{3} + x ( x ^{2} + 1 ) - 3 ( x ^{2} + 1 )}{3 x ^{2} ( x ^{2} + 1 )}

Expand

3 x^{2} (x^{2} + 1) : 3 x^{4} + 3 x^{2}

= \frac{3 x ^{3} + x ( x ^{2} + 1 ) - 3 ( x ^{2} + 1 )}{3 x ^{4} + 3 x ^{2}}

Expand

3 x^{3} + x (x^{2} + 1) - 3 (x^{2} + 1) : 4 x^{3} + x - 3 x^{2} - 3

= \frac{4 x ^{3} + x - 3 x ^{2} - 3}{3 x ^{4} + 3 x ^{2}}

\frac{( n + 3 ) ( n + 2 )}{( n + 1 ) ( n + 4 )}

\frac{( 4 - 16 x )}{( 5 - 4 x ) ( 2 x + 3 )}

\frac{cos ( x ^{3} )}{sin ( x ) ( 1 + sin ^{2} ( x ) )}

\frac{( h + 1 ) ( h + 2 ) ( 2 h + 3 )}{6}

U (t - π)

What is expand x/(x^2+1)+1/(3x)-1/(x^2) ?
The solution to expand x/(x^2+1)+1/(3x)-1/(x^2) is (4x^3+x-3x^2-3)/(3x^4+3x^2)