Integrand size = 20, antiderivative size = 73 \[ \int \frac {\left (a+\frac {b}{x}\right )^n x^m}{(c+d x)^2} \, dx=-\frac {\left (a+\frac {b}{x}\right )^n \left (1+\frac {b}{a x}\right )^{-n} x^{-1+m} \operatorname {AppellF1}\left (1-m,-n,2,2-m,-\frac {b}{a x},-\frac {c}{d x}\right )}{d^2 (1-m)} \]
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Time = 0.05 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {528, 511, 140, 138} \[ \int \frac {\left (a+\frac {b}{x}\right )^n x^m}{(c+d x)^2} \, dx=-\frac {x^{m-1} \left (a+\frac {b}{x}\right )^n \left (\frac {b}{a x}+1\right )^{-n} \operatorname {AppellF1}\left (1-m,-n,2,2-m,-\frac {b}{a x},-\frac {c}{d x}\right )}{d^2 (1-m)} \]
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Rule 138
Rule 140
Rule 511
Rule 528
Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (a+\frac {b}{x}\right )^n x^{-2+m}}{\left (d+\frac {c}{x}\right )^2} \, dx \\ & = -\left (\left (\left (\frac {1}{x}\right )^m x^m\right ) \text {Subst}\left (\int \frac {x^{-m} (a+b x)^n}{(d+c x)^2} \, dx,x,\frac {1}{x}\right )\right ) \\ & = -\left (\left (\left (a+\frac {b}{x}\right )^n \left (1+\frac {b}{a x}\right )^{-n} \left (\frac {1}{x}\right )^m x^m\right ) \text {Subst}\left (\int \frac {x^{-m} \left (1+\frac {b x}{a}\right )^n}{(d+c x)^2} \, dx,x,\frac {1}{x}\right )\right ) \\ & = -\frac {\left (a+\frac {b}{x}\right )^n \left (1+\frac {b}{a x}\right )^{-n} x^{-1+m} F_1\left (1-m;-n,2;2-m;-\frac {b}{a x},-\frac {c}{d x}\right )}{d^2 (1-m)} \\ \end{align*}
\[ \int \frac {\left (a+\frac {b}{x}\right )^n x^m}{(c+d x)^2} \, dx=\int \frac {\left (a+\frac {b}{x}\right )^n x^m}{(c+d x)^2} \, dx \]
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\[\int \frac {\left (a +\frac {b}{x}\right )^{n} x^{m}}{\left (d x +c \right )^{2}}d x\]
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\[ \int \frac {\left (a+\frac {b}{x}\right )^n x^m}{(c+d x)^2} \, dx=\int { \frac {{\left (a + \frac {b}{x}\right )}^{n} x^{m}}{{\left (d x + c\right )}^{2}} \,d x } \]
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\[ \int \frac {\left (a+\frac {b}{x}\right )^n x^m}{(c+d x)^2} \, dx=\int \frac {x^{m} \left (a + \frac {b}{x}\right )^{n}}{\left (c + d x\right )^{2}}\, dx \]
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\[ \int \frac {\left (a+\frac {b}{x}\right )^n x^m}{(c+d x)^2} \, dx=\int { \frac {{\left (a + \frac {b}{x}\right )}^{n} x^{m}}{{\left (d x + c\right )}^{2}} \,d x } \]
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\[ \int \frac {\left (a+\frac {b}{x}\right )^n x^m}{(c+d x)^2} \, dx=\int { \frac {{\left (a + \frac {b}{x}\right )}^{n} x^{m}}{{\left (d x + c\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+\frac {b}{x}\right )^n x^m}{(c+d x)^2} \, dx=\int \frac {x^m\,{\left (a+\frac {b}{x}\right )}^n}{{\left (c+d\,x\right )}^2} \,d x \]
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