Integrand size = 26, antiderivative size = 85 \[ \int \frac {(a+b x)^m (c+d x)^{1-m}}{(e+f x)^3} \, dx=\frac {(b c-a d)^2 (a+b x)^{1+m} (c+d x)^{-1-m} \operatorname {Hypergeometric2F1}\left (3,1+m,2+m,\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(b e-a f)^3 (1+m)} \]
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Time = 0.02 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {133} \[ \int \frac {(a+b x)^m (c+d x)^{1-m}}{(e+f x)^3} \, dx=\frac {(b c-a d)^2 (a+b x)^{m+1} (c+d x)^{-m-1} \operatorname {Hypergeometric2F1}\left (3,m+1,m+2,\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(m+1) (b e-a f)^3} \]
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Rule 133
Rubi steps \begin{align*} \text {integral}& = \frac {(b c-a d)^2 (a+b x)^{1+m} (c+d x)^{-1-m} \, _2F_1\left (3,1+m;2+m;\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(b e-a f)^3 (1+m)} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^m (c+d x)^{1-m}}{(e+f x)^3} \, dx=\frac {(b c-a d)^2 (a+b x)^{1+m} (c+d x)^{-1-m} \operatorname {Hypergeometric2F1}\left (3,1+m,2+m,\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(b e-a f)^3 (1+m)} \]
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\[\int \frac {\left (b x +a \right )^{m} \left (d x +c \right )^{1-m}}{\left (f x +e \right )^{3}}d x\]
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\[ \int \frac {(a+b x)^m (c+d x)^{1-m}}{(e+f x)^3} \, dx=\int { \frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m + 1}}{{\left (f x + e\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {(a+b x)^m (c+d x)^{1-m}}{(e+f x)^3} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+b x)^m (c+d x)^{1-m}}{(e+f x)^3} \, dx=\int { \frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m + 1}}{{\left (f x + e\right )}^{3}} \,d x } \]
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\[ \int \frac {(a+b x)^m (c+d x)^{1-m}}{(e+f x)^3} \, dx=\int { \frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m + 1}}{{\left (f x + e\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {(a+b x)^m (c+d x)^{1-m}}{(e+f x)^3} \, dx=\int \frac {{\left (a+b\,x\right )}^m\,{\left (c+d\,x\right )}^{1-m}}{{\left (e+f\,x\right )}^3} \,d x \]
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