This integral is solved by expanding the numerator, simplifying, and using both the power rule and the rule for integrating 1/x. Convert the radical √x in the numerator to a fractional exponent and convert the fraction 1/x^{3/2} to a negative exponent x^{–3/2}. Expand (1 + x^{1/2} )^{2} and multiply the result by x^{–3/2}. Then solve the three resulting integrals using the rule for integrating 1/x for one term and the power rule for the other two terms.

= ∫ (1 + x^{1/2} )^{2} x^{–3/2} dx  
= ∫ (1 + 2x^{1/2} + x) x^{–3/2} dx  
= ∫ (x^{–3/2} + 2x^{–1} + x^{–1/2}) dx  
= ∫ x^{–3/2} dx + 2 ∫ x^{–1} dx + ∫ x^{–1/2} dx  


= –2x^{–1/2} + 2 ln x + 2x^{1/2} + C 