Integrand size = 33, antiderivative size = 268 \[ \int (a+b x)^m (A+B x) (c+d x)^n (e+f x)^{-m-n} \, dx=\frac {(A b-a B) (a+b x)^{1+m} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^{-m-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{m+n} \operatorname {AppellF1}\left (1+m,-n,m+n,2+m,-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^2 (1+m)}+\frac {B (a+b x)^{2+m} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^{-m-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{m+n} \operatorname {AppellF1}\left (2+m,-n,m+n,3+m,-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^2 (2+m)} \]
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Time = 0.15 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {165, 145, 144, 143} \[ \int (a+b x)^m (A+B x) (c+d x)^n (e+f x)^{-m-n} \, dx=\frac {(A b-a B) (a+b x)^{m+1} (c+d x)^n (e+f x)^{-m-n} \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{m+n} \operatorname {AppellF1}\left (m+1,-n,m+n,m+2,-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^2 (m+1)}+\frac {B (a+b x)^{m+2} (c+d x)^n (e+f x)^{-m-n} \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{m+n} \operatorname {AppellF1}\left (m+2,-n,m+n,m+3,-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^2 (m+2)} \]
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Rule 143
Rule 144
Rule 145
Rule 165
Rubi steps \begin{align*} \text {integral}& = \frac {B \int (a+b x)^{1+m} (c+d x)^n (e+f x)^{-m-n} \, dx}{b}+\frac {(A b-a B) \int (a+b x)^m (c+d x)^n (e+f x)^{-m-n} \, dx}{b} \\ & = \frac {\left (B (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n}\right ) \int (a+b x)^{1+m} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n (e+f x)^{-m-n} \, dx}{b}+\frac {\left ((A b-a B) (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n}\right ) \int (a+b x)^m \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n (e+f x)^{-m-n} \, dx}{b} \\ & = \frac {\left (B (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^{-m-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{m+n}\right ) \int (a+b x)^{1+m} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n \left (\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}\right )^{-m-n} \, dx}{b}+\frac {\left ((A b-a B) (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^{-m-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{m+n}\right ) \int (a+b x)^m \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n \left (\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}\right )^{-m-n} \, dx}{b} \\ & = \frac {(A b-a B) (a+b x)^{1+m} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^{-m-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{m+n} F_1\left (1+m;-n,m+n;2+m;-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^2 (1+m)}+\frac {B (a+b x)^{2+m} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^{-m-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{m+n} F_1\left (2+m;-n,m+n;3+m;-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^2 (2+m)} \\ \end{align*}
\[ \int (a+b x)^m (A+B x) (c+d x)^n (e+f x)^{-m-n} \, dx=\int (a+b x)^m (A+B x) (c+d x)^n (e+f x)^{-m-n} \, dx \]
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\[\int \left (b x +a \right )^{m} \left (B x +A \right ) \left (d x +c \right )^{n} \left (f x +e \right )^{-n -m}d x\]
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\[ \int (a+b x)^m (A+B x) (c+d x)^n (e+f x)^{-m-n} \, dx=\int { {\left (B x + A\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{n} {\left (f x + e\right )}^{-m - n} \,d x } \]
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Timed out. \[ \int (a+b x)^m (A+B x) (c+d x)^n (e+f x)^{-m-n} \, dx=\text {Timed out} \]
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\[ \int (a+b x)^m (A+B x) (c+d x)^n (e+f x)^{-m-n} \, dx=\int { {\left (B x + A\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{n} {\left (f x + e\right )}^{-m - n} \,d x } \]
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\[ \int (a+b x)^m (A+B x) (c+d x)^n (e+f x)^{-m-n} \, dx=\int { {\left (B x + A\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{n} {\left (f x + e\right )}^{-m - n} \,d x } \]
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Timed out. \[ \int (a+b x)^m (A+B x) (c+d x)^n (e+f x)^{-m-n} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (a+b\,x\right )}^m\,{\left (c+d\,x\right )}^n}{{\left (e+f\,x\right )}^{m+n}} \,d x \]
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