Solve 5/(x+1)+(frac{x{x^2+4})+(8/x^3)}

$$\frac{ 5 }{ (x+1)+( \frac{ x }{ { x }^{ 2 } +4 } )+( \frac{ 8 }{ { x }^{ 3 } } ) }$$

Evaluate

$$\frac{5\left(x^{2}+4\right)x^{3}}{x^{6}+x^{5}+5x^{4}+4x^{3}+8x^{2}+32}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply x+1 times \frac{x^{2}+4}{x^{2}+4}.

$$\frac{5}{\frac{\left(x+1\right)\left(x^{2}+4\right)}{x^{2}+4}+\frac{x}{x^{2}+4}+\frac{8}{x^{3}}}$$

Since \frac{\left(x+1\right)\left(x^{2}+4\right)}{x^{2}+4} and \frac{x}{x^{2}+4} have the same denominator, add them by adding their numerators.

$$\frac{5}{\frac{\left(x+1\right)\left(x^{2}+4\right)+x}{x^{2}+4}+\frac{8}{x^{3}}}$$

Do the multiplications in \left(x+1\right)\left(x^{2}+4\right)+x.

$$\frac{5}{\frac{x^{3}+4x+x^{2}+4+x}{x^{2}+4}+\frac{8}{x^{3}}}$$

Combine like terms in x^{3}+4x+x^{2}+4+x.

$$\frac{5}{\frac{x^{3}+5x+x^{2}+4}{x^{2}+4}+\frac{8}{x^{3}}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{2}+4 and x^{3} is \left(x^{2}+4\right)x^{3}. Multiply \frac{x^{3}+5x+x^{2}+4}{x^{2}+4} times \frac{x^{3}}{x^{3}}. Multiply \frac{8}{x^{3}} times \frac{x^{2}+4}{x^{2}+4}.

$$\frac{5}{\frac{\left(x^{3}+5x+x^{2}+4\right)x^{3}}{\left(x^{2}+4\right)x^{3}}+\frac{8\left(x^{2}+4\right)}{\left(x^{2}+4\right)x^{3}}}$$

Since \frac{\left(x^{3}+5x+x^{2}+4\right)x^{3}}{\left(x^{2}+4\right)x^{3}} and \frac{8\left(x^{2}+4\right)}{\left(x^{2}+4\right)x^{3}} have the same denominator, add them by adding their numerators.

$$\frac{5}{\frac{\left(x^{3}+5x+x^{2}+4\right)x^{3}+8\left(x^{2}+4\right)}{\left(x^{2}+4\right)x^{3}}}$$

Do the multiplications in \left(x^{3}+5x+x^{2}+4\right)x^{3}+8\left(x^{2}+4\right).

$$\frac{5}{\frac{x^{6}+5x^{4}+x^{5}+4x^{3}+8x^{2}+32}{\left(x^{2}+4\right)x^{3}}}$$

Divide 5 by \frac{x^{6}+5x^{4}+x^{5}+4x^{3}+8x^{2}+32}{\left(x^{2}+4\right)x^{3}} by multiplying 5 by the reciprocal of \frac{x^{6}+5x^{4}+x^{5}+4x^{3}+8x^{2}+32}{\left(x^{2}+4\right)x^{3}}.

$$\frac{5\left(x^{2}+4\right)x^{3}}{x^{6}+5x^{4}+x^{5}+4x^{3}+8x^{2}+32}$$

Use the distributive property to multiply 5 by x^{2}+4.

$$\frac{\left(5x^{2}+20\right)x^{3}}{x^{6}+5x^{4}+x^{5}+4x^{3}+8x^{2}+32}$$

Use the distributive property to multiply 5x^{2}+20 by x^{3}.

$$\frac{5x^{5}+20x^{3}}{x^{6}+5x^{4}+x^{5}+4x^{3}+8x^{2}+32}$$

Differentiate w.r.t. x

$$5\times \left(\frac{x}{x^{6}+x^{5}+5x^{4}+4x^{3}+8x^{2}+32}\right)^{2}\left(384+192x^{2}+4x^{4}-7x^{6}-x^{8}\right)$$