Solve (x/x+1+x+1/x^2-1)divx^2+1/x^2+x

$$( \frac { x } { x + 1 } + \frac { x + 1 } { x ^ { 2 } - 1 } ) \div \frac { x ^ { 2 } + 1 } { x ^ { 2 } + x }$$

Evaluate

$$\frac{x}{x-1}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{x+1}{x^{2}-1}.

$$\frac{\frac{x}{x+1}+\frac{x+1}{\left(x-1\right)\left(x+1\right)}}{\frac{x^{2}+1}{x^{2}+x}}$$

Cancel out x+1 in both numerator and denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+1 and x-1 is \left(x-1\right)\left(x+1\right). Multiply \frac{x}{x+1} times \frac{x-1}{x-1}. Multiply \frac{1}{x-1} times \frac{x+1}{x+1}.

$$\frac{\frac{x\left(x-1\right)}{\left(x-1\right)\left(x+1\right)}+\frac{x+1}{\left(x-1\right)\left(x+1\right)}}{\frac{x^{2}+1}{x^{2}+x}}$$

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

$$\frac{\frac{x\left(x-1\right)+x+1}{\left(x-1\right)\left(x+1\right)}}{\frac{x^{2}+1}{x^{2}+x}}$$

Do the multiplications in x\left(x-1\right)+x+1.

$$\frac{\frac{x^{2}-x+x+1}{\left(x-1\right)\left(x+1\right)}}{\frac{x^{2}+1}{x^{2}+x}}$$

Combine like terms in x^{2}-x+x+1.

$$\frac{\frac{x^{2}+1}{\left(x-1\right)\left(x+1\right)}}{\frac{x^{2}+1}{x^{2}+x}}$$

Divide \frac{x^{2}+1}{\left(x-1\right)\left(x+1\right)} by \frac{x^{2}+1}{x^{2}+x} by multiplying \frac{x^{2}+1}{\left(x-1\right)\left(x+1\right)} by the reciprocal of \frac{x^{2}+1}{x^{2}+x}.

$$\frac{\left(x^{2}+1\right)\left(x^{2}+x\right)}{\left(x-1\right)\left(x+1\right)\left(x^{2}+1\right)}$$

Cancel out x^{2}+1 in both numerator and denominator.

$$\frac{x^{2}+x}{\left(x-1\right)\left(x+1\right)}$$

Factor the expressions that are not already factored.

$$\frac{x\left(x+1\right)}{\left(x-1\right)\left(x+1\right)}$$

Cancel out x+1 in both numerator and denominator.

$$\frac{x}{x-1}$$

Expand

$$\frac{x}{x-1}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{x+1}{x^{2}-1}.

$$\frac{\frac{x}{x+1}+\frac{x+1}{\left(x-1\right)\left(x+1\right)}}{\frac{x^{2}+1}{x^{2}+x}}$$

Cancel out x+1 in both numerator and denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+1 and x-1 is \left(x-1\right)\left(x+1\right). Multiply \frac{x}{x+1} times \frac{x-1}{x-1}. Multiply \frac{1}{x-1} times \frac{x+1}{x+1}.

$$\frac{\frac{x\left(x-1\right)}{\left(x-1\right)\left(x+1\right)}+\frac{x+1}{\left(x-1\right)\left(x+1\right)}}{\frac{x^{2}+1}{x^{2}+x}}$$

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

$$\frac{\frac{x\left(x-1\right)+x+1}{\left(x-1\right)\left(x+1\right)}}{\frac{x^{2}+1}{x^{2}+x}}$$

Do the multiplications in x\left(x-1\right)+x+1.

$$\frac{\frac{x^{2}-x+x+1}{\left(x-1\right)\left(x+1\right)}}{\frac{x^{2}+1}{x^{2}+x}}$$

Combine like terms in x^{2}-x+x+1.

$$\frac{\frac{x^{2}+1}{\left(x-1\right)\left(x+1\right)}}{\frac{x^{2}+1}{x^{2}+x}}$$

Divide \frac{x^{2}+1}{\left(x-1\right)\left(x+1\right)} by \frac{x^{2}+1}{x^{2}+x} by multiplying \frac{x^{2}+1}{\left(x-1\right)\left(x+1\right)} by the reciprocal of \frac{x^{2}+1}{x^{2}+x}.

$$\frac{\left(x^{2}+1\right)\left(x^{2}+x\right)}{\left(x-1\right)\left(x+1\right)\left(x^{2}+1\right)}$$

Cancel out x^{2}+1 in both numerator and denominator.

$$\frac{x^{2}+x}{\left(x-1\right)\left(x+1\right)}$$

Factor the expressions that are not already factored.

$$\frac{x\left(x+1\right)}{\left(x-1\right)\left(x+1\right)}$$

Cancel out x+1 in both numerator and denominator.

$$\frac{x}{x-1}$$