Integrand size = 26, antiderivative size = 432 \[ \int \frac {(a+b x)^m (c+d x)^{-2-m}}{(e+f x)^3} \, dx=\frac {d \left (a^2 d^2 f^2 \left (6+5 m+m^2\right )+b^2 \left (2 d^2 e^2+5 c d e f (1+m)-c^2 f^2 \left (1-m^2\right )\right )-a b d f \left (d e (9+5 m)+c f \left (3+5 m+2 m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-1-m}}{2 (b c-a d) (b e-a f)^2 (d e-c f)^3 (1+m)}-\frac {f (a+b x)^{1+m} (c+d x)^{-1-m}}{2 (b e-a f) (d e-c f) (e+f x)^2}-\frac {f (b (4 d e-c f (1-m))-a d f (3+m)) (a+b x)^{1+m} (c+d x)^{-1-m}}{2 (b e-a f)^2 (d e-c f)^2 (e+f x)}-\frac {f \left (2 a b d f (2+m) (3 d e+c f m)-b^2 \left (6 d^2 e^2+6 c d e f m-c^2 f^2 (1-m) m\right )-a^2 d^2 f^2 \left (6+5 m+m^2\right )\right ) (a+b x)^m (c+d x)^{-m} \operatorname {Hypergeometric2F1}\left (1,-m,1-m,\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{2 (b e-a f)^2 (d e-c f)^4 m} \]
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Time = 0.35 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {105, 156, 160, 12, 133} \[ \int \frac {(a+b x)^m (c+d x)^{-2-m}}{(e+f x)^3} \, dx=-\frac {f (a+b x)^m (c+d x)^{-m} \left (-a^2 d^2 f^2 \left (m^2+5 m+6\right )+2 a b d f (m+2) (c f m+3 d e)-\left (b^2 \left (-c^2 f^2 (1-m) m+6 c d e f m+6 d^2 e^2\right )\right )\right ) \operatorname {Hypergeometric2F1}\left (1,-m,1-m,\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{2 m (b e-a f)^2 (d e-c f)^4}+\frac {d (a+b x)^{m+1} (c+d x)^{-m-1} \left (a^2 d^2 f^2 \left (m^2+5 m+6\right )-a b d f \left (c f \left (2 m^2+5 m+3\right )+d e (5 m+9)\right )+b^2 \left (-c^2 f^2 \left (1-m^2\right )+5 c d e f (m+1)+2 d^2 e^2\right )\right )}{2 (m+1) (b c-a d) (b e-a f)^2 (d e-c f)^3}-\frac {f (a+b x)^{m+1} (c+d x)^{-m-1} (-a d f (m+3)-b c f (1-m)+4 b d e)}{2 (e+f x) (b e-a f)^2 (d e-c f)^2}-\frac {f (a+b x)^{m+1} (c+d x)^{-m-1}}{2 (e+f x)^2 (b e-a f) (d e-c f)} \]
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Rule 12
Rule 105
Rule 133
Rule 156
Rule 160
Rubi steps \begin{align*} \text {integral}& = -\frac {f (a+b x)^{1+m} (c+d x)^{-1-m}}{2 (b e-a f) (d e-c f) (e+f x)^2}-\frac {\int \frac {(a+b x)^m (c+d x)^{-2-m} (-2 b d e+b c f (1-m)+a d f (3+m)+2 b d f x)}{(e+f x)^2} \, dx}{2 (b e-a f) (d e-c f)} \\ & = -\frac {f (a+b x)^{1+m} (c+d x)^{-1-m}}{2 (b e-a f) (d e-c f) (e+f x)^2}-\frac {f (4 b d e-b c f (1-m)-a d f (3+m)) (a+b x)^{1+m} (c+d x)^{-1-m}}{2 (b e-a f)^2 (d e-c f)^2 (e+f x)}+\frac {\int \frac {(a+b x)^m (c+d x)^{-2-m} \left (a^2 d^2 f^2 \left (6+5 m+m^2\right )+b^2 \left (2 d^2 e^2-c^2 f^2 (1-m) m+c d e f (1+5 m)\right )-a b d f (2 c f m (2+m)+d e (9+5 m))-b d f (4 b d e-b c f (1-m)-a d f (3+m)) x\right )}{e+f x} \, dx}{2 (b e-a f)^2 (d e-c f)^2} \\ & = \frac {d \left (a^2 d^2 f^2 \left (6+5 m+m^2\right )+b^2 \left (2 d^2 e^2+5 c d e f (1+m)-c^2 f^2 \left (1-m^2\right )\right )-a b d f \left (d e (9+5 m)+c f \left (3+5 m+2 m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-1-m}}{2 (b c-a d) (b e-a f)^2 (d e-c f)^3 (1+m)}-\frac {f (a+b x)^{1+m} (c+d x)^{-1-m}}{2 (b e-a f) (d e-c f) (e+f x)^2}-\frac {f (4 b d e-b c f (1-m)-a d f (3+m)) (a+b x)^{1+m} (c+d x)^{-1-m}}{2 (b e-a f)^2 (d e-c f)^2 (e+f x)}+\frac {\int \frac {(b c-a d) f (1+m) \left (2 a b d f (2+m) (3 d e+c f m)-b^2 \left (6 d^2 e^2+6 c d e f m-c^2 f^2 (1-m) m\right )-a^2 d^2 f^2 \left (6+5 m+m^2\right )\right ) (a+b x)^m (c+d x)^{-1-m}}{e+f x} \, dx}{2 (b c-a d) (b e-a f)^2 (d e-c f)^3 (1+m)} \\ & = \frac {d \left (a^2 d^2 f^2 \left (6+5 m+m^2\right )+b^2 \left (2 d^2 e^2+5 c d e f (1+m)-c^2 f^2 \left (1-m^2\right )\right )-a b d f \left (d e (9+5 m)+c f \left (3+5 m+2 m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-1-m}}{2 (b c-a d) (b e-a f)^2 (d e-c f)^3 (1+m)}-\frac {f (a+b x)^{1+m} (c+d x)^{-1-m}}{2 (b e-a f) (d e-c f) (e+f x)^2}-\frac {f (4 b d e-b c f (1-m)-a d f (3+m)) (a+b x)^{1+m} (c+d x)^{-1-m}}{2 (b e-a f)^2 (d e-c f)^2 (e+f x)}+\frac {\left (f \left (2 a b d f (2+m) (3 d e+c f m)-b^2 \left (6 d^2 e^2+6 c d e f m-c^2 f^2 (1-m) m\right )-a^2 d^2 f^2 \left (6+5 m+m^2\right )\right )\right ) \int \frac {(a+b x)^m (c+d x)^{-1-m}}{e+f x} \, dx}{2 (b e-a f)^2 (d e-c f)^3} \\ & = \frac {d \left (a^2 d^2 f^2 \left (6+5 m+m^2\right )+b^2 \left (2 d^2 e^2+5 c d e f (1+m)-c^2 f^2 \left (1-m^2\right )\right )-a b d f \left (d e (9+5 m)+c f \left (3+5 m+2 m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-1-m}}{2 (b c-a d) (b e-a f)^2 (d e-c f)^3 (1+m)}-\frac {f (a+b x)^{1+m} (c+d x)^{-1-m}}{2 (b e-a f) (d e-c f) (e+f x)^2}-\frac {f (4 b d e-b c f (1-m)-a d f (3+m)) (a+b x)^{1+m} (c+d x)^{-1-m}}{2 (b e-a f)^2 (d e-c f)^2 (e+f x)}-\frac {f \left (2 a b d f (2+m) (3 d e+c f m)-b^2 \left (6 d^2 e^2+6 c d e f m-c^2 f^2 (1-m) m\right )-a^2 d^2 f^2 \left (6+5 m+m^2\right )\right ) (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 )}{2 (b e-a f)^2 (d e-c f)^4 m} \\ \end{align*}
Time = 0.53 (sec) , antiderivative size = 374, normalized size of antiderivative = 0.87 \[ \int \frac {(a+b x)^m (c+d x)^{-2-m}}{(e+f x)^3} \, dx=-\frac {(a+b x)^{1+m} (c+d x)^{-m} \left (f (b e-a f)^2 (d e-c f) (2 b d e+b c f (1+m)-a d f (3+m))+\frac {2 d (b e-a f)^3 (d e-c f)^2}{c+d x}+(e+f x) \left (f (b e-a f) \left (a^2 d^2 f^2 \left (6+5 m+m^2\right )+b^2 \left (2 d^2 e^2+5 c d e f (1+m)+c^2 f^2 \left (-1+m^2\right )\right )-a b d f \left (d e (9+5 m)+c f \left (3+5 m+2 m^2\right )\right )\right )-\frac {(b c-a d) f \left (-2 a b d f (2+m) (3 d e+c f m)+b^2 \left (6 d^2 e^2+6 c d e f m+c^2 f^2 (-1+m) m\right )+a^2 d^2 f^2 \left (6+5 m+m^2\right )\right ) (e+f x) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{c+d x}\right )\right )}{2 (b c-a d) (b e-a f)^3 (d e-c f)^2 (-d e+c f) (1+m) (e+f x)^2} \]
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\[\int \frac {\left (b x +a \right )^{m} \left (d x +c \right )^{-2-m}}{\left (f x +e \right )^{3}}d x\]
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\[ \int \frac {(a+b x)^m (c+d x)^{-2-m}}{(e+f x)^3} \, dx=\int { \frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 2}}{{\left (f x + e\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {(a+b x)^m (c+d x)^{-2-m}}{(e+f x)^3} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+b x)^m (c+d x)^{-2-m}}{(e+f x)^3} \, dx=\int { \frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 2}}{{\left (f x + e\right )}^{3}} \,d x } \]
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\[ \int \frac {(a+b x)^m (c+d x)^{-2-m}}{(e+f x)^3} \, dx=\int { \frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 2}}{{\left (f x + e\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {(a+b x)^m (c+d x)^{-2-m}}{(e+f x)^3} \, dx=\int \frac {{\left (a+b\,x\right )}^m}{{\left (e+f\,x\right )}^3\,{\left (c+d\,x\right )}^{m+2}} \,d x \]
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