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\frac{1}{a+b}+\frac{b\left(a+b\right)}{a+b}+\frac{b}{a\left(a+b\right)}

To add or subtract expressions, expand them to make their denominators the same. Multiply b times \frac{a+b}{a+b}.

\frac{1+b\left(a+b\right)}{a+b}+\frac{b}{a\left(a+b\right)}

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

\frac{1+ba+b^{2}}{a+b}+\frac{b}{a\left(a+b\right)}

Do the multiplications in 1+b\left(a+b\right).

\frac{\left(1+ba+b^{2}\right)a}{a\left(a+b\right)}+\frac{b}{a\left(a+b\right)}

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a+b and a\left(a+b\right) is a\left(a+b\right). Multiply \frac{1+ba+b^{2}}{a+b} times \frac{a}{a}.

\frac{\left(1+ba+b^{2}\right)a+b}{a\left(a+b\right)}

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

\frac{a+ba^{2}+b^{2}a+b}{a\left(a+b\right)}

Do the multiplications in \left(1+ba+b^{2}\right)a+b.

\frac{\left(a+b\right)\left(ab+1\right)}{a\left(a+b\right)}

Factor the expressions that are not already factored in \frac{a+ba^{2}+b^{2}a+b}{a\left(a+b\right)}.

\frac{ab+1}{a}

Cancel out a+b in both numerator and denominator.