A.10. Integrals with $a + bu$ — formulas and resources

\int \frac{u\,du}{a + bu} = \frac{1}{b^2}\left(a + bu - a\,\ln|a + bu|\right) + C

\int \frac{u^2\,du}{a + bu} = \frac{1}{2b^3}\left[(a+bu)^2 - 4a(a+bu) + 2a^2\,\ln|a+bu|\right] + C

\int \frac{du}{u(a + bu)} = \frac{1}{a}\,\ln\left|\frac{u}{a + bu}\right| + C

\int \frac{du}{u^2(a + bu)} = -\frac{1}{au} + \frac{b}{a^2}\,\ln\left|\frac{a + bu}{u}\right| + C

\int \frac{u\,du}{(a + bu)^2} = \frac{a}{b^2(a + bu)} + \frac{1}{b^2}\,\ln|a + bu| + C

\int \frac{du}{u(a + bu)^2} = \frac{1}{a(a + bu)} - \frac{1}{a^2}\,\ln\left|\frac{a + bu}{u}\right| + C

\int \frac{u^2\,du}{(a + bu)^2} = \frac{1}{b^3}\left(a + bu - \frac{a^2}{a + bu} - 2a\,\ln|a + bu|\right) + C

\sqrt{a + bu}

\int u\,\sqrt{a + bu}\,du = \frac{2}{15b^2}(3bu - 2a)(a + bu)^{3/2} + C

\int \frac{u\,du}{\sqrt{a + bu}} = \frac{2}{3b^2}(bu - 2a)\,\sqrt{a + bu} + C

\int \frac{u^2\,du}{\sqrt{a + bu}} = \frac{2}{15b^3}\left(8a^2 + 3b^2 u^2 - 4abu\right)\sqrt{a + bu} + C

\int \frac{du}{u\,\sqrt{a + bu}} = \frac{1}{\sqrt{a}}\,\ln\left|\frac{\sqrt{a + bu} - \sqrt{a}}{\sqrt{a + bu} + \sqrt{a}}\right| + C, \quad \text{if } a > 0