Integrand size = 20, antiderivative size = 63 \[ \int \frac {(b x)^m (c+d x)^n}{e+f x} \, dx=\frac {(b x)^{1+m} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} \operatorname {AppellF1}\left (1+m,-n,1,2+m,-\frac {d x}{c},-\frac {f x}{e}\right )}{b e (1+m)} \]
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Time = 0.02 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {140, 138} \[ \int \frac {(b x)^m (c+d x)^n}{e+f x} \, dx=\frac {(b x)^{m+1} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \operatorname {AppellF1}\left (m+1,-n,1,m+2,-\frac {d x}{c},-\frac {f x}{e}\right )}{b e (m+1)} \]
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Rule 138
Rule 140
Rubi steps \begin{align*} \text {integral}& = \left ((c+d x)^n \left (1+\frac {d x}{c}\right )^{-n}\right ) \int \frac {(b x)^m \left (1+\frac {d x}{c}\right )^n}{e+f x} \, dx \\ & = \frac {(b x)^{1+m} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} F_1\left (1+m;-n,1;2+m;-\frac {d x}{c},-\frac {f x}{e}\right )}{b e (1+m)} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.95 \[ \int \frac {(b x)^m (c+d x)^n}{e+f x} \, dx=\frac {x (b x)^m (c+d x)^n \left (\frac {c+d x}{c}\right )^{-n} \operatorname {AppellF1}\left (1+m,-n,1,2+m,-\frac {d x}{c},-\frac {f x}{e}\right )}{e (1+m)} \]
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\[\int \frac {\left (b x \right )^{m} \left (d x +c \right )^{n}}{f x +e}d x\]
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\[ \int \frac {(b x)^m (c+d x)^n}{e+f x} \, dx=\int { \frac {\left (b x\right )^{m} {\left (d x + c\right )}^{n}}{f x + e} \,d x } \]
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Exception generated. \[ \int \frac {(b x)^m (c+d x)^n}{e+f x} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
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\[ \int \frac {(b x)^m (c+d x)^n}{e+f x} \, dx=\int { \frac {\left (b x\right )^{m} {\left (d x + c\right )}^{n}}{f x + e} \,d x } \]
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\[ \int \frac {(b x)^m (c+d x)^n}{e+f x} \, dx=\int { \frac {\left (b x\right )^{m} {\left (d x + c\right )}^{n}}{f x + e} \,d x } \]
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Timed out. \[ \int \frac {(b x)^m (c+d x)^n}{e+f x} \, dx=\int \frac {{\left (b\,x\right )}^m\,{\left (c+d\,x\right )}^n}{e+f\,x} \,d x \]
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