Integrand size = 22, antiderivative size = 135 \[ \int (a+b x)^m (c+d x)^{-m} (e+f x) \, dx=\frac {f (a+b x)^{1+m} (c+d x)^{1-m}}{2 b d}-\frac {(a d f (1-m)-b (2 d e-c f (1+m))) (a+b x)^{1+m} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \operatorname {Hypergeometric2F1}\left (m,1+m,2+m,-\frac {d (a+b x)}{b c-a d}\right )}{2 b^2 d (1+m)} \]
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Time = 0.04 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {81, 72, 71} \[ \int (a+b x)^m (c+d x)^{-m} (e+f x) \, dx=\frac {(a+b x)^{m+1} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m (-a d f (1-m)-b c f (m+1)+2 b d e) \operatorname {Hypergeometric2F1}\left (m,m+1,m+2,-\frac {d (a+b x)}{b c-a d}\right )}{2 b^2 d (m+1)}+\frac {f (a+b x)^{m+1} (c+d x)^{1-m}}{2 b d} \]
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Rule 71
Rule 72
Rule 81
Rubi steps \begin{align*} \text {integral}& = \frac {f (a+b x)^{1+m} (c+d x)^{1-m}}{2 b d}+\frac {(2 b d e-f (a d (1-m)+b c (1+m))) \int (a+b x)^m (c+d x)^{-m} \, dx}{2 b d} \\ & = \frac {f (a+b x)^{1+m} (c+d x)^{1-m}}{2 b d}+\frac {\left ((2 b d e-f (a d (1-m)+b c (1+m))) (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m\right ) \int (a+b x)^m \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{-m} \, dx}{2 b d} \\ & = \frac {f (a+b x)^{1+m} (c+d x)^{1-m}}{2 b d}+\frac {(2 b d e-a d f (1-m)-b c f (1+m)) (a+b x)^{1+m} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,1+m;2+m;-\frac {d (a+b x)}{b c-a d}\right )}{2 b^2 d (1+m)} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.81 \[ \int (a+b x)^m (c+d x)^{-m} (e+f x) \, dx=\frac {(a+b x)^{1+m} (c+d x)^{-m} \left (b f (c+d x)-\frac {(-2 b d e-a d f (-1+m)+b c f (1+m)) \left (\frac {b (c+d x)}{b c-a d}\right )^m \operatorname {Hypergeometric2F1}\left (m,1+m,2+m,\frac {d (a+b x)}{-b c+a d}\right )}{1+m}\right )}{2 b^2 d} \]
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\[\int \left (b x +a \right )^{m} \left (f x +e \right ) \left (d x +c \right )^{-m}d x\]
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\[ \int (a+b x)^m (c+d x)^{-m} (e+f x) \, dx=\int { \frac {{\left (f x + e\right )} {\left (b x + a\right )}^{m}}{{\left (d x + c\right )}^{m}} \,d x } \]
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Exception generated. \[ \int (a+b x)^m (c+d x)^{-m} (e+f x) \, dx=\text {Exception raised: HeuristicGCDFailed} \]
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\[ \int (a+b x)^m (c+d x)^{-m} (e+f x) \, dx=\int { \frac {{\left (f x + e\right )} {\left (b x + a\right )}^{m}}{{\left (d x + c\right )}^{m}} \,d x } \]
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\[ \int (a+b x)^m (c+d x)^{-m} (e+f x) \, dx=\int { \frac {{\left (f x + e\right )} {\left (b x + a\right )}^{m}}{{\left (d x + c\right )}^{m}} \,d x } \]
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Timed out. \[ \int (a+b x)^m (c+d x)^{-m} (e+f x) \, dx=\int \frac {\left (e+f\,x\right )\,{\left (a+b\,x\right )}^m}{{\left (c+d\,x\right )}^m} \,d x \]
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