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A.10. Integrals with $a + b u$

\int \frac{u d u}{a + b u} = \frac{1}{b ^{2}} (a + b u - a ln ∣ a + b u ∣) + C

\int \frac{u ^{2} d u}{a + b u} = \frac{1}{2 b ^{3}} [(a + b u)^{2} - 4 a (a + b u) + 2 a^{2} ln ∣ a + b u ∣] + C

\int \frac{d u}{u ( a + b u )} = \frac{1}{a} ln \frac{u}{a + b u} + C

\int \frac{d u}{u ^{2} ( a + b u )} = - \frac{1}{a u} + \frac{b}{a ^{2}} ln \frac{a + b u}{u} + C

\int \frac{u d u}{( a + b u ) ^{2}} = \frac{a}{b ^{2} ( a + b u )} + \frac{1}{b ^{2}} ln ∣ a + b u ∣ + C

\int \frac{d u}{u ( a + b u ) ^{2}} = \frac{1}{a ( a + b u )} - \frac{1}{a ^{2}} ln \frac{a + b u}{u} + C

\int \frac{u ^{2} d u}{( a + b u ) ^{2}} = \frac{1}{b ^{3}} (a + b u - \frac{a ^{2}}{a + b u} - 2 a ln ∣ a + b u ∣) + C

a + b u

\int u a + b u d u = \frac{2}{15 b ^{2}} (3 b u - 2 a) (a + b u)^{3/2} + C

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

\int \frac{u ^{2} d u}{a + b u} = \frac{2}{15 b ^{3}} (8 a^{2} + 3 b^{2} u^{2} - 4 ab u) a + b u + C

\int \frac{d u}{u a + b u} = \frac{1}{a} ln \frac{a + b u - a}{a + b u + a} + C, si a > 0