Integrand size = 20, antiderivative size = 81 \[ \int (b x)^m (c+d x)^n (e+f x)^p \, dx=\frac {(b x)^{1+m} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} (e+f x)^p \left (1+\frac {f x}{e}\right )^{-p} \operatorname {AppellF1}\left (1+m,-n,-p,2+m,-\frac {d x}{c},-\frac {f x}{e}\right )}{b (1+m)} \]
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Time = 0.03 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {140, 138} \[ \int (b x)^m (c+d x)^n (e+f x)^p \, dx=\frac {(b x)^{m+1} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} (e+f x)^p \left (\frac {f x}{e}+1\right )^{-p} \operatorname {AppellF1}\left (m+1,-n,-p,m+2,-\frac {d x}{c},-\frac {f x}{e}\right )}{b (m+1)} \]
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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 (b x)^m \left (1+\frac {d x}{c}\right )^n (e+f x)^p \, dx \\ & = \left ((c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} (e+f x)^p \left (1+\frac {f x}{e}\right )^{-p}\right ) \int (b x)^m \left (1+\frac {d x}{c}\right )^n \left (1+\frac {f x}{e}\right )^p \, dx \\ & = \frac {(b x)^{1+m} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} (e+f x)^p \left (1+\frac {f x}{e}\right )^{-p} F_1\left (1+m;-n,-p;2+m;-\frac {d x}{c},-\frac {f x}{e}\right )}{b (1+m)} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.98 \[ \int (b x)^m (c+d x)^n (e+f x)^p \, dx=\frac {x (b x)^m (c+d x)^n \left (\frac {c+d x}{c}\right )^{-n} (e+f x)^p \left (\frac {e+f x}{e}\right )^{-p} \operatorname {AppellF1}\left (1+m,-n,-p,2+m,-\frac {d x}{c},-\frac {f x}{e}\right )}{1+m} \]
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\[\int \left (b x \right )^{m} \left (d x +c \right )^{n} \left (f x +e \right )^{p}d x\]
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\[ \int (b x)^m (c+d x)^n (e+f x)^p \, dx=\int { \left (b x\right )^{m} {\left (d x + c\right )}^{n} {\left (f x + e\right )}^{p} \,d x } \]
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Timed out. \[ \int (b x)^m (c+d x)^n (e+f x)^p \, dx=\text {Timed out} \]
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\[ \int (b x)^m (c+d x)^n (e+f x)^p \, dx=\int { \left (b x\right )^{m} {\left (d x + c\right )}^{n} {\left (f x + e\right )}^{p} \,d x } \]
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\[ \int (b x)^m (c+d x)^n (e+f x)^p \, dx=\int { \left (b x\right )^{m} {\left (d x + c\right )}^{n} {\left (f x + e\right )}^{p} \,d x } \]
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Timed out. \[ \int (b x)^m (c+d x)^n (e+f x)^p \, dx=\int {\left (e+f\,x\right )}^p\,{\left (b\,x\right )}^m\,{\left (c+d\,x\right )}^n \,d x \]
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