Integrand size = 26, antiderivative size = 557 \[ \int \frac {(a+b x)^m (c+d x)^{-5-m}}{e+f x} \, dx=\frac {d (a+b x)^{1+m} (c+d x)^{-4-m}}{(b c-a d) (d e-c f) (4+m)}+\frac {d (a d f (4+m)+b (3 d e-c f (7+m))) (a+b x)^{1+m} (c+d x)^{-3-m}}{(b c-a d)^2 (d e-c f)^2 (3+m) (4+m)}+\frac {d \left (a^2 d^2 f^2 \left (12+7 m+m^2\right )+2 a b d f (4+m) (d e-c f (4+m))+b^2 \left (6 d^2 e^2-2 c d e f (10+m)+c^2 f^2 \left (26+9 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-2-m}}{(b c-a d)^3 (d e-c f)^3 (2+m) (3+m) (4+m)}+\frac {d \left (a^3 d^3 f^3 \left (24+26 m+9 m^2+m^3\right )+a^2 b d^2 f^2 \left (12+7 m+m^2\right ) (d e-c f (7+3 m))+a b^2 d f (4+m) \left (2 d^2 e^2-2 c d e f (5+m)+c^2 f^2 \left (26+17 m+3 m^2\right )\right )+b^3 \left (6 d^3 e^3-2 c d^2 e^2 f (13+m)+c^2 d e f^2 \left (46+11 m+m^2\right )-c^3 f^3 \left (50+35 m+10 m^2+m^3\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-1-m}}{(b c-a d)^4 (d e-c f)^4 (1+m) (2+m) (3+m) (4+m)}-\frac {f^4 (a+b x)^m (c+d x)^{-m} \operatorname {Hypergeometric2F1}\left (1,-m,1-m,\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{(d e-c f)^5 m} \]
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Time = 0.72 (sec) , antiderivative size = 556, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {136, 160, 12, 133} \[ \int \frac {(a+b x)^m (c+d x)^{-5-m}}{e+f x} \, dx=\frac {d (a+b x)^{m+1} (c+d x)^{-m-2} \left (a^2 d^2 f^2 \left (m^2+7 m+12\right )+2 a b d f (m+4) (d e-c f (m+4))+b^2 \left (c^2 f^2 \left (m^2+9 m+26\right )-2 c d e f (m+10)+6 d^2 e^2\right )\right )}{(m+2) (m+3) (m+4) (b c-a d)^3 (d e-c f)^3}+\frac {d (a+b x)^{m+1} (c+d x)^{-m-1} \left (a^3 d^3 f^3 \left (m^3+9 m^2+26 m+24\right )+a^2 b d^2 f^2 \left (m^2+7 m+12\right ) (d e-c f (3 m+7))+a b^2 d f (m+4) \left (c^2 f^2 \left (3 m^2+17 m+26\right )-2 c d e f (m+5)+2 d^2 e^2\right )+b^3 \left (-c^3 f^3 \left (m^3+10 m^2+35 m+50\right )+c^2 d e f^2 \left (m^2+11 m+46\right )-2 c d^2 e^2 f (m+13)+6 d^3 e^3\right )\right )}{(m+1) (m+2) (m+3) (m+4) (b c-a d)^4 (d e-c f)^4}-\frac {f^4 (a+b x)^m (c+d x)^{-m} \operatorname {Hypergeometric2F1}\left (1,-m,1-m,\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{m (d e-c f)^5}+\frac {d (a+b x)^{m+1} (c+d x)^{-m-4}}{(m+4) (b c-a d) (d e-c f)}+\frac {d (a+b x)^{m+1} (c+d x)^{-m-3} (a d f (m+4)-b c f (m+7)+3 b d e)}{(m+3) (m+4) (b c-a d)^2 (d e-c f)^2} \]
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Rule 12
Rule 133
Rule 136
Rule 160
Rubi steps \begin{align*} \text {integral}& = \frac {d (a+b x)^{1+m} (c+d x)^{-4-m}}{(b c-a d) (d e-c f) (4+m)}+\frac {\int \frac {(a+b x)^m (c+d x)^{-4-m} (3 b d e-b c f (4+m)+a d f (4+m)+3 b d f x)}{e+f x} \, dx}{(b c-a d) (d e-c f) (4+m)} \\ & = \frac {d (a+b x)^{1+m} (c+d x)^{-4-m}}{(b c-a d) (d e-c f) (4+m)}+\frac {d (3 b d e+a d f (4+m)-b c f (7+m)) (a+b x)^{1+m} (c+d x)^{-3-m}}{(b c-a d)^2 (d e-c f)^2 (3+m) (4+m)}+\frac {\int \frac {(a+b x)^m (c+d x)^{-3-m} \left (a^2 d^2 f^2 \left (12+7 m+m^2\right )+2 a b d f (4+m) (d e-c f (3+m))+b^2 \left (6 d^2 e^2-2 c d e f (7+m)+c^2 f^2 \left (12+7 m+m^2\right )\right )+2 b d f (3 b d e+a d f (4+m)-b c f (7+m)) x\right )}{e+f x} \, dx}{(b c-a d)^2 (d e-c f)^2 (3+m) (4+m)} \\ & = \frac {d (a+b x)^{1+m} (c+d x)^{-4-m}}{(b c-a d) (d e-c f) (4+m)}+\frac {d (3 b d e+a d f (4+m)-b c f (7+m)) (a+b x)^{1+m} (c+d x)^{-3-m}}{(b c-a d)^2 (d e-c f)^2 (3+m) (4+m)}+\frac {d \left (a^2 d^2 f^2 \left (12+7 m+m^2\right )+2 a b d f (4+m) (d e-c f (4+m))+b^2 \left (6 d^2 e^2-2 c d e f (10+m)+c^2 f^2 \left (26+9 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-2-m}}{(b c-a d)^3 (d e-c f)^3 (2+m) (3+m) (4+m)}+\frac {\int \frac {(a+b x)^m (c+d x)^{-2-m} \left (a^3 d^3 f^3 \left (24+26 m+9 m^2+m^3\right )+a^2 b d^2 f^2 \left (12+7 m+m^2\right ) (d e-3 c f (2+m))+a b^2 d f (4+m) \left (2 d^2 e^2-2 c d e f (4+m)+3 c^2 f^2 \left (6+5 m+m^2\right )\right )+b^3 \left (6 d^3 e^3-2 c d^2 e^2 f (10+m)+c^2 d e f^2 \left (26+9 m+m^2\right )-c^3 f^3 \left (24+26 m+9 m^2+m^3\right )\right )+b d f \left (a^2 d^2 f^2 \left (12+7 m+m^2\right )+2 a b d f (4+m) (d e-c f (4+m))+b^2 \left (6 d^2 e^2-2 c d e f (10+m)+c^2 f^2 \left (26+9 m+m^2\right )\right )\right ) x\right )}{e+f x} \, dx}{(b c-a d)^3 (d e-c f)^3 (2+m) (3+m) (4+m)} \\ & = \frac {d (a+b x)^{1+m} (c+d x)^{-4-m}}{(b c-a d) (d e-c f) (4+m)}+\frac {d (3 b d e+a d f (4+m)-b c f (7+m)) (a+b x)^{1+m} (c+d x)^{-3-m}}{(b c-a d)^2 (d e-c f)^2 (3+m) (4+m)}+\frac {d \left (a^2 d^2 f^2 \left (12+7 m+m^2\right )+2 a b d f (4+m) (d e-c f (4+m))+b^2 \left (6 d^2 e^2-2 c d e f (10+m)+c^2 f^2 \left (26+9 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-2-m}}{(b c-a d)^3 (d e-c f)^3 (2+m) (3+m) (4+m)}+\frac {d \left (a^3 d^3 f^3 \left (24+26 m+9 m^2+m^3\right )+a^2 b d^2 f^2 \left (12+7 m+m^2\right ) (d e-c f (7+3 m))+a b^2 d f (4+m) \left (2 d^2 e^2-2 c d e f (5+m)+c^2 f^2 \left (26+17 m+3 m^2\right )\right )+b^3 \left (6 d^3 e^3-2 c d^2 e^2 f (13+m)+c^2 d e f^2 \left (46+11 m+m^2\right )-c^3 f^3 \left (50+35 m+10 m^2+m^3\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-1-m}}{(b c-a d)^4 (d e-c f)^4 (1+m) (2+m) (3+m) (4+m)}+\frac {\int \frac {(b c-a d)^4 f^4 (1+m) (2+m) (3+m) (4+m) (a+b x)^m (c+d x)^{-1-m}}{e+f x} \, dx}{(b c-a d)^4 (d e-c f)^4 (1+m) (2+m) (3+m) (4+m)} \\ & = \frac {d (a+b x)^{1+m} (c+d x)^{-4-m}}{(b c-a d) (d e-c f) (4+m)}+\frac {d (3 b d e+a d f (4+m)-b c f (7+m)) (a+b x)^{1+m} (c+d x)^{-3-m}}{(b c-a d)^2 (d e-c f)^2 (3+m) (4+m)}+\frac {d \left (a^2 d^2 f^2 \left (12+7 m+m^2\right )+2 a b d f (4+m) (d e-c f (4+m))+b^2 \left (6 d^2 e^2-2 c d e f (10+m)+c^2 f^2 \left (26+9 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-2-m}}{(b c-a d)^3 (d e-c f)^3 (2+m) (3+m) (4+m)}+\frac {d \left (a^3 d^3 f^3 \left (24+26 m+9 m^2+m^3\right )+a^2 b d^2 f^2 \left (12+7 m+m^2\right ) (d e-c f (7+3 m))+a b^2 d f (4+m) \left (2 d^2 e^2-2 c d e f (5+m)+c^2 f^2 \left (26+17 m+3 m^2\right )\right )+b^3 \left (6 d^3 e^3-2 c d^2 e^2 f (13+m)+c^2 d e f^2 \left (46+11 m+m^2\right )-c^3 f^3 \left (50+35 m+10 m^2+m^3\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-1-m}}{(b c-a d)^4 (d e-c f)^4 (1+m) (2+m) (3+m) (4+m)}+\frac {f^4 \int \frac {(a+b x)^m (c+d x)^{-1-m}}{e+f x} \, dx}{(d e-c f)^4} \\ & = \frac {d (a+b x)^{1+m} (c+d x)^{-4-m}}{(b c-a d) (d e-c f) (4+m)}+\frac {d (3 b d e+a d f (4+m)-b c f (7+m)) (a+b x)^{1+m} (c+d x)^{-3-m}}{(b c-a d)^2 (d e-c f)^2 (3+m) (4+m)}+\frac {d \left (a^2 d^2 f^2 \left (12+7 m+m^2\right )+2 a b d f (4+m) (d e-c f (4+m))+b^2 \left (6 d^2 e^2-2 c d e f (10+m)+c^2 f^2 \left (26+9 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-2-m}}{(b c-a d)^3 (d e-c f)^3 (2+m) (3+m) (4+m)}+\frac {d \left (a^3 d^3 f^3 \left (24+26 m+9 m^2+m^3\right )+a^2 b d^2 f^2 \left (12+7 m+m^2\right ) (d e-c f (7+3 m))+a b^2 d f (4+m) \left (2 d^2 e^2-2 c d e f (5+m)+c^2 f^2 \left (26+17 m+3 m^2\right )\right )+b^3 \left (6 d^3 e^3-2 c d^2 e^2 f (13+m)+c^2 d e f^2 \left (46+11 m+m^2\right )-c^3 f^3 \left (50+35 m+10 m^2+m^3\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-1-m}}{(b c-a d)^4 (d e-c f)^4 (1+m) (2+m) (3+m) (4+m)}-\frac {f^4 (a+b x)^m (c+d x)^{-m} \, _2F_1\left (1,-m;1-m;\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{(d e-c f)^5 m} \\ \end{align*}
Time = 1.36 (sec) , antiderivative size = 525, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b x)^m (c+d x)^{-5-m}}{e+f x} \, dx=-\frac {(a+b x)^{1+m} (c+d x)^{-4-m} \left (d-\frac {d (3 b d e+a d f (4+m)-b c f (7+m)) (c+d x)}{(b c-a d) (-d e+c f) (3+m)}-\frac {(c+d x)^2 \left (d (b c-a d) (b e-a f) (d e-c f) (1+m)^2 \left (-a^2 d^2 f^2 \left (12+7 m+m^2\right )+2 a b d f (4+m) (-d e+c f (4+m))-b^2 \left (6 d^2 e^2-2 c d e f (10+m)+c^2 f^2 \left (26+9 m+m^2\right )\right )\right )+(c+d x) \left (-d (b e-a f) (1+m) \left (a^3 d^3 f^3 \left (24+26 m+9 m^2+m^3\right )+a^2 b d^2 f^2 \left (12+7 m+m^2\right ) (d e-c f (7+3 m))+a b^2 d f (4+m) \left (2 d^2 e^2-2 c d e f (5+m)+c^2 f^2 \left (26+17 m+3 m^2\right )\right )+b^3 \left (6 d^3 e^3-2 c d^2 e^2 f (13+m)+c^2 d e f^2 \left (46+11 m+m^2\right )-c^3 f^3 \left (50+35 m+10 m^2+m^3\right )\right )\right )-(b c-a d)^4 f^4 \left (24+50 m+35 m^2+10 m^3+m^4\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )\right )\right )}{(b c-a d)^3 (b e-a f) (d e-c f)^3 (1+m)^2 (2+m) (3+m)}\right )}{(b c-a d) (-d e+c f) (4+m)} \]
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\[\int \frac {\left (b x +a \right )^{m} \left (d x +c \right )^{-5-m}}{f x +e}d x\]
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\[ \int \frac {(a+b x)^m (c+d x)^{-5-m}}{e+f x} \, dx=\int { \frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 5}}{f x + e} \,d x } \]
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Timed out. \[ \int \frac {(a+b x)^m (c+d x)^{-5-m}}{e+f x} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+b x)^m (c+d x)^{-5-m}}{e+f x} \, dx=\int { \frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 5}}{f x + e} \,d x } \]
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\[ \int \frac {(a+b x)^m (c+d x)^{-5-m}}{e+f x} \, dx=\int { \frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 5}}{f x + e} \,d x } \]
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Timed out. \[ \int \frac {(a+b x)^m (c+d x)^{-5-m}}{e+f x} \, dx=\int \frac {{\left (a+b\,x\right )}^m}{\left (e+f\,x\right )\,{\left (c+d\,x\right )}^{m+5}} \,d x \]
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