Fórmulas que contienen a + bu

$$\int \frac{udu}{a+bu}=\frac{1}{b^2}(a+bu-a\:ln \:|a+bu|)+C$$

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

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

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

$$\int \frac{udu}{(a+bu)^2}=\frac{a}{b^2(a+bu)}+\frac{1}{b^2}ln|a+bu|+C$$

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

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

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

$$\int \frac{udu}{\sqrt{a+bu}}=\frac{2}{3b^2}(bu-2a)\sqrt{a+bu}+C$$

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

$$\int \frac{du}{u\sqrt{a+bu}}= \left\{ \begin{array}{ll}\frac{1}{\sqrt{a}}ln|\frac{\sqrt{a+bu}-\sqrt{a}}{\sqrt{a+bu}+\sqrt{a}}|+C \hspace{2em} (a>0) \\\ \frac{2}{\sqrt{-a}}tan^{-1}\sqrt{\frac{a+bu}{-a}}+C \hspace{2em} (a<0)\end{array}\right.$$

$$\int \frac{\sqrt{a+bu}}{u}du=2\sqrt{a+bu}+a \int \frac{du}{u\sqrt{a+bu}}$$

$$\int \frac{\sqrt{a+bu}}{u^2}du=-\frac{\sqrt{a+bu}}{u}+\frac{b}{2} \int \frac{du}{u\sqrt{a+bu}}$$

$$\int u^n \sqrt{a+bu}du=\frac{2u^n(a+bu)^{\frac{3}{2}}}{b(2n+3)}-\frac{2na}{b(2n+3)}\int \frac{u^{n-1}}{\sqrt{a+bu}}du$$

$$\int \frac{u^ndu}{\sqrt{a+bu}}=\frac{2u^n\sqrt{a+bu}}{b(2n+1)}-\frac{2na}{b(2n+1)} \int \frac{u^{n-1}du}{\sqrt{a+bu}}$$

$$\int \frac{du}{u^n \sqrt{a+bu}}=-\frac{\sqrt{a+bu}}{a(n-1)u^{n-1}}-\frac{b(2n-3)}{2a(n-1)}\int\frac{du}{u^{n-1}\sqrt{a+bu}}$$