What is the integral of (u^2)/((a+bu)^2) ?

The integral of (u^2)/((a+bu)^2) is -(u^2)/(b(a+bu))+2/(b^3)(a+bu-aln|a+bu|)+C

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integral of (u^2)/((a+bu)^2)

\int \frac{u ^{2}}{( a + b u ) ^{2}} d u

- \frac{u ^{2}}{b ( a + b u )} + \frac{2}{b ^{3}} (a + b u - a ln ∣ a + b u ∣) + C

Solution steps

\int \frac{u ^{2}}{( a + b u ) ^{2}} d u

Apply Integration By Parts

: - \frac{u ^{2}}{b ( a + b u )} - \int - \frac{2 u}{b ( a + b u )} d u

= - \frac{u ^{2}}{b ( a + b u )} - \int - \frac{2 u}{b ( a + b u )} d u

\int - \frac{2 u}{b ( a + b u )} d u = - \frac{2}{b ^{3}} (a + b u - a ln ∣ a + b u ∣)

= - \frac{u ^{2}}{b ( a + b u )} - (- \frac{2}{b ^{3}} (a + b u - a ln ∣ a + b u ∣))

Simplify

= - \frac{u ^{2}}{b ( a + b u )} + \frac{2}{b ^{3}} (a + b u - a ln ∣ a + b u ∣)

Add a constant to the solution

= - \frac{u ^{2}}{b ( a + b u )} + \frac{2}{b ^{3}} (a + b u - a ln ∣ a + b u ∣) + C

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What is the integral of (u^2)/((a+bu)^2) ?
The integral of (u^2)/((a+bu)^2) is -(u^2)/(b(a+bu))+2/(b^3)(a+bu-aln|a+bu|)+C