Indefinite Integrals with ax+b and px+q

 $\int \frac {d x}{(ax+b)(px+q)} = \frac {1}{bp-aq} \ln \left ( \frac {px+q}{ax+b} \right )+C$ $\int \frac {x d x}{(ax+b)(px+q)} = \frac {1}{bp-aq} \left \{ \frac{b}{a} \ln { (ax+b)} - \frac{q}{p} \ln{(px+q)} \right \}+C$ $\int \frac { d x}{(ax+b)^2(px+q)} = \frac {1}{bp-aq} \left \{ \frac {1}{ax+b} + \frac {p}{bp- aq} \ln \left ( \frac {px+q}{ax +b} \right ) \right \}+C$ $\int \frac {x d x}{(ax+b)^2(px+q)} = \frac {1}{bp-aq} \left \{ \frac {q}{bp- aq} \ln \left ( \frac {ax+b}{px +q} \right ) - \frac {b}{a(ax+b)} \right \}+C$ $\int \frac {x^2 d x}{(ax+b)^2(px+q)} = \frac {b^2}{(bp -aq)a^2(ax+b)} + \frac {1}{(bp-aq)^2} \left \{ \frac {q^2}{p} \ln (px+q) + \frac {b(bp-2aq)}{a^2} \ln (ax+b) \right \}+C$ $\int \frac { d x }{(ax+b)^m (px+q)^n } = \frac {-1}{(n-1)(bp-aq)} \left \{ \frac {1}{(ax+b)^{m-1}(pz+q)^{n-1} } + a(m + n -2) \ \int \frac {d x}{(ax+b)^m(px+q)^{n-1} } \right \}$ $\int \frac {ax+b}{px+q} d x = \frac {ax}{p} + \frac {bp -aq}{p^2} \ln (px+q)+C$ $\int \frac {(ax+b)^m}{(px+q)^n} d x = \begin{cases} \frac {-1}{(n-1)(bp-aq)} \left \{ \frac {(ax+b)^{m+1}}{(px+q)^{n-1}} + (n-m-2)a \int \frac {(ax+b)^m}{(px+q)^{n-1}} d x \right \} \\ \frac {-1}{(n-m-1)p} \left \{ \frac {(ax+b)^m}{(px+q)^{n-1}} + m(bp-aq) \int \frac{(ax+b)^{m-1}} {(px+q)^n} d x \right \} \\ \frac{-1}{(n-1)p} \left \{ \frac{(ax+b)^m}{(px+q)^{n-1}} - ma \int \frac{(ax+b)^{m-1}}{(px+q)^{n-1}} d x \right \} \end{cases}$ $\int \frac {px+q}{\sqrt {ax+b}} d x = \frac {2(apx+3aq-2bp)}{3a^2} \sqrt {ax+b}+C$ $\int \frac {d x}{(px+q) \sqrt {ax+b}} = \begin{cases} \frac{1}{ \sqrt {bp-aq} \sqrt {p} } \ln \left ( \frac {\sqrt{p(ax+b)} -\sqrt {bp-aq} }{\sqrt{p(ax+b)} + \sqrt {bp-aq} } \right )+C \\ \frac {2}{\sqrt {aq-bp} \sqrt {p}} \arctan \sqrt{ \frac {p(ax+b)}{aq-bp} }+C \end{cases}$ $\int \frac {\sqrt {ax+b}}{px+q} \ d x = \begin{cases} \frac{2 \sqrt{ax+b}}{p} \ + \ \frac {\sqrt {bp-aq}}{p \sqrt{p}} \ln \left ( \frac {\sqrt {p(ax+b)} - \sqrt {bp-aq}}{\sqrt {p(ax+b)} + \sqrt {bp-aq}} \right )+C \\ \frac {2 \sqrt {ax+b}}{p} \ - \ \frac {2 \sqrt{aq-bp}}{p \sqrt{p}} \arctan \sqrt { \frac {p(ax+b)}{aq-bp}} +C\\ \end{cases}$ $\int (px+q)^n \sqrt {ax+b} \ d x = \frac{2(px+q)^{n+1} \sqrt {ax+b}}{(2n+3)p} \ + \ \frac {bp-aq}{(2n+3)p} \int \frac {(px+q)^n}{\sqrt {ax+b}} d x$ $\int \frac {d x}{ (px+q)^n \sqrt {ax+b}} = \frac {\sqrt {ax+b}}{(n-1)(aq-bp)(px+q)^{n-1}} + \frac {(2n-3)a}{2(n-1)(aq-bp)} \int \frac {d x }{(px+q)^{n-1} \sqrt{ax+b}}$ $\int \frac {(px+q)^n}{\sqrt {ax+b}} d x = \frac {2(px+q)^n \sqrt{ax+b}}{)2n+1)a} + \frac {2n(aq-bp)}{(2n+1)a} \int \frac {(px+q)^{n-1} d x}{\sqrt{ax+b}}$ $\int \frac {\sqrt{ax+b}}{(px+q)^n} d x = \frac {- \sqrt {ax+b}}{(n-1)p(px+q)^n-1} + \frac{a}{2(n-1)p} \int \frac {d x}{(px+q)^{n-1} \sqrt {ax+b}}$ $\int \frac {d x}{\sqrt {(ax+b)(px+q)}} = \begin{cases} \frac{2}{\sqrt{ap}} \ln { \left ( \sqrt {a(px+q)}+ \sqrt{p(ax+b)} \right )} +C\\ \frac {2}{\sqrt { -ap}} \arctan \ \sqrt { \frac {-p(ax+b)} {a(px+q)} } +C \\ \end{cases}$ $\int \frac {x d x}{\sqrt {(ax+b)(px+q)}} = \frac {\sqrt{(ax+b)(px+q)}}{ap} - \frac{bp+aq}{2ap} \int \frac {d x}{\sqrt{(ax+b)(px+q)}}$ $\int \sqrt {(ax+b)(px+q)} dx = \frac {2apx + bp +aq}{4ap} \sqrt {(ax+b)(px+q)} - \frac {(bp-aq)^2}{8ap} \int \frac{d x}{\sqrt{(ax+b)(px+q)}}$ $\int \sqrt {\frac {px+q}{ax+b}} dx = \frac {\sqrt{(ax+b)(px+q)}}{a} + \frac {aq-bp}{2a} \int \frac {d x}{ \sqrt {(ax+b)(px+q)}}$ $\int \frac {d x}{(px+q)\sqrt{(ax+b)(px+q)}} = \frac {2\sqrt{ax+b}}{(aq-bp)\sqrt{px+q}}+C$