Integrand size = 20, antiderivative size = 79 \[ \int x^m (a+b x)^{1+n} (c+d x)^n \, dx=\frac {a x^{1+m} (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} \operatorname {AppellF1}\left (1+m,-1-n,-n,2+m,-\frac {b x}{a},-\frac {d x}{c}\right )}{1+m} \]
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Time = 0.03 (sec) , antiderivative size = 79, 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} \[ \int x^m (a+b x)^{1+n} (c+d x)^n \, dx=\frac {a x^{m+1} (a+b x)^n \left (\frac {b x}{a}+1\right )^{-n} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \operatorname {AppellF1}\left (m+1,-n-1,-n,m+2,-\frac {b x}{a},-\frac {d x}{c}\right )}{m+1} \]
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Rule 138
Rule 140
Rubi steps \begin{align*} \text {integral}& = \left (a (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n}\right ) \int x^m \left (1+\frac {b x}{a}\right )^{1+n} (c+d x)^n \, dx \\ & = \left (a (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n}\right ) \int x^m \left (1+\frac {b x}{a}\right )^{1+n} \left (1+\frac {d x}{c}\right )^n \, dx \\ & = \frac {a x^{1+m} (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} F_1\left (1+m;-1-n,-n;2+m;-\frac {b x}{a},-\frac {d x}{c}\right )}{1+m} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.52 \[ \int x^m (a+b x)^{1+n} (c+d x)^n \, dx=\frac {x^{1+m} (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} \left (a (2+m) \operatorname {AppellF1}\left (1+m,-n,-n,2+m,-\frac {b x}{a},-\frac {d x}{c}\right )+b (1+m) x \operatorname {AppellF1}\left (2+m,-n,-n,3+m,-\frac {b x}{a},-\frac {d x}{c}\right )\right )}{(1+m) (2+m)} \]
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\[\int x^{m} \left (b x +a \right )^{1+n} \left (d x +c \right )^{n}d x\]
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\[ \int x^m (a+b x)^{1+n} (c+d x)^n \, dx=\int { {\left (b x + a\right )}^{n + 1} {\left (d x + c\right )}^{n} x^{m} \,d x } \]
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Exception generated. \[ \int x^m (a+b x)^{1+n} (c+d x)^n \, dx=\text {Exception raised: HeuristicGCDFailed} \]
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\[ \int x^m (a+b x)^{1+n} (c+d x)^n \, dx=\int { {\left (b x + a\right )}^{n + 1} {\left (d x + c\right )}^{n} x^{m} \,d x } \]
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\[ \int x^m (a+b x)^{1+n} (c+d x)^n \, dx=\int { {\left (b x + a\right )}^{n + 1} {\left (d x + c\right )}^{n} x^{m} \,d x } \]
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Timed out. \[ \int x^m (a+b x)^{1+n} (c+d x)^n \, dx=\int x^m\,{\left (a+b\,x\right )}^{n+1}\,{\left (c+d\,x\right )}^n \,d x \]
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