$\int\frac{\text{x}}{\sqrt{\text{x}+\text{a}}-\sqrt{\text{x}+\text{b}}}\text{dx}$
$=\int\frac{\text{x}}{\sqrt{\text{x}+\text{a}}-\sqrt{\text{x}+\text{b}}}\times\frac{\sqrt{\text{x}+\text{a}}+\sqrt{\text{x}+\text{b}}}{\sqrt{\text{x}+\text{a}}+\sqrt{\text{x}+\text{b}}}\text{dx}$
$=\int\frac{\text{x}(\sqrt{\text{x}+\text{a}}+\sqrt{\text{x}+\text{b}})}{(\sqrt{\text{x}+\text{a}})^2-(\sqrt{\text{x}+\text{b}})^2}\text{dx}$
$=\int\frac{\text{x}(\sqrt{\text{x}+\text{a}}+\sqrt{\text{x}+\text{b}})}{\text{x}+\text{a}-\text{x}-\text{b}}\text{dx}$
$=\frac{1}{\text{a}-\text{b}}\int\text{x}(\sqrt{\text{x}+\text{a}}+\sqrt{\text{x}+\text{b}})\text{dx}$
$=\frac{1}{\text{a}-\text{b}}\big[\int\text{x}(\sqrt{\text{x}+\text{a}})\text{dx}+\int\text{x}(\sqrt{\text{x}+\text{b}})\text{dx}\big]$
$=\frac{1}{\text{a}-\text{b}}\big[\int(\text{x}+\text{a}-\text{a})(\sqrt{\text{x}+\text{a}})\text{dx}+\int(\text{x}+\text{b}-\text{b})(\sqrt{x+\text{b}})\text{dx}$
$=\frac{1}{\text{a}-\text{b}}\big[\int(\text{x}+\text{a})(\sqrt{\text{x}+\text{a}})\text{dx}-\text{a}\int(\sqrt{\text{x}+\text{a}})\text{dx}\\+\int(\text{x}+\text{b})(\sqrt{\text{x}+\text{b}})\text{dx}-\text{b}\int(\sqrt{\text{x}+\text{b}})\text{dx}\big]$
$=\frac{1}{\text{a}-\text{b}}\big[\int(\text{x}+\text{a})^\frac{3}{2}\text{dx}-\text{a}\int(\text{x}+\text{a})^\frac{1}{2}\text{dx}+\int(\text{x}+\text{b})^\frac{3}{2}\text{dx}-\text{b}\int(\text{x}+\text{b})^\frac{1}{2}\text{dx}\big]$
$=\frac{1}{\text{a}-\text{b}}\Big[\frac{(\text{x}+\text{a})^\frac{5}{2}}{\frac{5}{2}}-\text{a}\frac{(\text{x}+\text{a})^\frac{3}{2}}{\frac{3}{2}}+\frac{(\text{x}+\text{b})^\frac{5}{2}}{\frac{5}{2}}-\text{b}\frac{(\text{x}+\text{b})^\frac{3}{2}}{\frac{3}{2}}+\text{c}$ where, c is an arbitrary constant.
$=\frac{1}{\text{a}-\text{b}}\Big[\frac{2}{5}(\text{x}+\text{a})^\frac{5}{2}-\frac{2\text{a}}{3}(\text{x}+\text{a})^\frac{3}{2}+\frac{2}{5}(\text{x}+\text{b})^\frac{5}{2}-\frac{2\text{b}}{3}(\text{x}+\text{b})^\frac{3}{2}\Big]+\text{c}$ where, c is an arbitrary constant.
Hence, $\int\frac{\text{x}}{\sqrt{\text{x}+\text{a}}-\sqrt{\text{x}+\text{b}}}\text{dx}=$ $\frac{1}{\text{a}-\text{b}}\Big[\frac{2}{5}(\text{x}+\text{a})^\frac{5}{2}-\frac{2\text{a}}{3}(\text{x}+\text{a})^\frac{3}{2}+\frac{2}{5}(\text{x}+\text{b})^\frac{5}{2}-\frac{2\text{b}}{3}(\text{x}+\text{b})^\frac{3}{2}\Big]+\text{c}$ where, c is an arbitrary constant.
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