Matematica

Con $\small \sqrt{ax+b}$ e $\small px+q$

1.: $\displaystyle \int\displaystyle \frac{px+q}{\displaystyle \sqrt{ax+b}}\,dx=\displaystyle \frac{2(apx+3aq-2bp)}{3a^{\displaystyle2 }}\displaystyle \sqrt{ax+b}$

2.: $\displaystyle \int\displaystyle \frac{dx}{(px+q)\displaystyle \sqrt{ax+b}}=\left\{\begin{array}{l} \displaystyle \frac{1}{\displaystyle \sqrt{bp-aq}{\displaystyle \sqrt{p}}}\ln\left(\displaystyle \frac{\displaystyle \sqrt{p(ax+b)}-\displaystyle \sqrt{bp-aq}}{\displaystyle \sqrt{p(ax+b)}+\displaystyle \sqrt{bp-aq}}\right)\\ \\ \displaystyle \frac{2}{\displaystyle \sqrt{aq-bp}\displaystyle \sqrt{p}}\tan^{\displaystyle-1}\displaystyle \sqrt{\displaystyle \frac{p(ax+b)}{aq-bp}} \end{array} \right.$

3.: $\displaystyle \int\displaystyle \frac{\displaystyle \sqrt{ax+b}}{px+q}\,dx=\left\{\begin{array}{l} \displaystyle \frac{2\displaystyle \sqrt{ax+b}}{p}+\displaystyle \frac{\displaystyle \sqrt{bp-aq}}{p\displaystyle \sqrt{p}}\ln\left(\displaystyle \frac{\displaystyle \sqrt{p(ax+b)}- \displaystyle \sqrt{bp-aq}}{\displaystyle \sqrt{p(ax+b)}+\displaystyle \sqrt{bp-aq}}\right)\\ \\ \displaystyle \frac{2\displaystyle \sqrt{ax+b}}{p}-\displaystyle \frac{2\displaystyle \sqrt{aq-bp}}{p\displaystyle \sqrt{p}}\tan^{\displaystyle-1}\displaystyle \sqrt{\displaystyle \frac{p(ax+b)}{aq-bp}} \end{array} \right.$

4.: $\small \displaystyle \int(px+q)^{\displaystyle n}\displaystyle \sqrt{ax+b}\,dx=\displaystyle \frac{2(px+q)^{\displaystyle n+1}\displaystyle \sqrt{ax+b}}{(2n+3)p}+\displaystyle \frac{bp-aq}{(2n+3)p}\displaystyle \int\displaystyle \frac{(px+q)^{\displaystyle n}}{\displaystyle \sqrt{ax+b}}\,dx$

5.: $\small \displaystyle \int\displaystyle \frac{(px+q)^{\displaystyle n}}{\displaystyle \sqrt{ax+b}}\,dx=\displaystyle \frac{2(px+q)^{\displaystyle n}\displaystyle \sqrt{ax+b}}{(2n+1)a}+\displaystyle \frac{2n(aq-bp)}{(2n+1)a}\displaystyle \int\displaystyle \frac{(px+q)^{\displaystyle n-1}\,dx}{\displaystyle \sqrt{ax+b}}$

6.: $\small \displaystyle \int\displaystyle \frac{\displaystyle \sqrt{ax+b}}{(px+q)^{\displaystyle n}}\,dx=\displaystyle \frac{-\displaystyle \sqrt{ax+b}}{(n-1)p(px+q)^{\displaystyle n-1}}+\displaystyle \frac{a}{2(n-1)p}\displaystyle \int\displaystyle \frac{dx}{(px+q)^{\displaystyle n-1}\displaystyle \sqrt{ax+b}}$

7.: $\displaystyle \begin{array}{lcl} \displaystyle \int\displaystyle \frac{dx}{(px+q)^{\displaystyle n}\displaystyle \sqrt{ax+b}}&=&\displaystyle \frac{\displaystyle \sqrt{ax+b}}{(n-1)(aq-bp)(px+q)^{\displaystyle n-1}}\\ \\ && \left. +\displaystyle \frac{(2n-3)a}{2(n-1)(aq-bp)}\displaystyle \int\displaystyle \frac{dx}{(px+q)^{\displaystyle n-1}\displaystyle \sqrt{ax+b}}\right.\end{array}$