Integrand size = 30, antiderivative size = 124 \[ \int (a+b x)^m (c+d x)^{-m-n} (e+f x)^{-2+n} \, dx=\frac {(a+b x)^{1+m} (c+d x)^{-m-n} \left (\frac {(b e-a f) (c+d x)}{(b c-a d) (e+f x)}\right )^{m+n} (e+f x)^{-1+n} \operatorname {Hypergeometric2F1}\left (1+m,m+n,2+m,-\frac {(d e-c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{(b e-a f) (1+m)} \]
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Time = 0.02 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {134} \[ \int (a+b x)^m (c+d x)^{-m-n} (e+f x)^{-2+n} \, dx=\frac {(a+b x)^{m+1} (e+f x)^{n-1} (c+d x)^{-m-n} \left (\frac {(c+d x) (b e-a f)}{(e+f x) (b c-a d)}\right )^{m+n} \operatorname {Hypergeometric2F1}\left (m+1,m+n,m+2,-\frac {(d e-c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{(m+1) (b e-a f)} \]
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Rule 134
Rubi steps \begin{align*} \text {integral}& = \frac {(a+b x)^{1+m} (c+d x)^{-m-n} \left (\frac {(b e-a f) (c+d x)}{(b c-a d) (e+f x)}\right )^{m+n} (e+f x)^{-1+n} \, _2F_1\left (1+m,m+n;2+m;-\frac {(d e-c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{(b e-a f) (1+m)} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.99 \[ \int (a+b x)^m (c+d x)^{-m-n} (e+f x)^{-2+n} \, dx=\frac {(a+b x)^{1+m} (c+d x)^{-m-n} \left (\frac {(b e-a f) (c+d x)}{(b c-a d) (e+f x)}\right )^{m+n} (e+f x)^{-1+n} \operatorname {Hypergeometric2F1}\left (1+m,m+n,2+m,\frac {(-d e+c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{(b e-a f) (1+m)} \]
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\[\int \left (b x +a \right )^{m} \left (d x +c \right )^{-n -m} \left (f x +e \right )^{-2+n}d x\]
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\[ \int (a+b x)^m (c+d x)^{-m-n} (e+f x)^{-2+n} \, dx=\int { {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - n} {\left (f x + e\right )}^{n - 2} \,d x } \]
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Timed out. \[ \int (a+b x)^m (c+d x)^{-m-n} (e+f x)^{-2+n} \, dx=\text {Timed out} \]
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\[ \int (a+b x)^m (c+d x)^{-m-n} (e+f x)^{-2+n} \, dx=\int { {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - n} {\left (f x + e\right )}^{n - 2} \,d x } \]
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\[ \int (a+b x)^m (c+d x)^{-m-n} (e+f x)^{-2+n} \, dx=\int { {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - n} {\left (f x + e\right )}^{n - 2} \,d x } \]
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Timed out. \[ \int (a+b x)^m (c+d x)^{-m-n} (e+f x)^{-2+n} \, dx=\int \frac {{\left (e+f\,x\right )}^{n-2}\,{\left (a+b\,x\right )}^m}{{\left (c+d\,x\right )}^{m+n}} \,d x \]
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