Integrand size = 24, antiderivative size = 309 \[ \int \frac {(a+b x)^m (c+d x)^{-m}}{(e+f x)^4} \, dx=-\frac {f (a+b x)^{1+m} (c+d x)^{1-m}}{3 (b e-a f) (d e-c f) (e+f x)^3}-\frac {f (b (4 d e-c f (2-m))-a d f (2+m)) (a+b x)^{1+m} (c+d x)^{1-m}}{6 (b e-a f)^2 (d e-c f)^2 (e+f x)^2}-\frac {(b c-a d) \left (2 a b d f (3 d e-c f (1-m)) (1+m)-a^2 d^2 f^2 \left (2+3 m+m^2\right )-b^2 \left (6 d^2 e^2-6 c d e f (1-m)+c^2 f^2 \left (2-3 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-1-m} \operatorname {Hypergeometric2F1}\left (2,1+m,2+m,\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{6 (b e-a f)^4 (d e-c f)^2 (1+m)} \]
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Time = 0.20 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {105, 156, 12, 133} \[ \int \frac {(a+b x)^m (c+d x)^{-m}}{(e+f x)^4} \, dx=-\frac {(b c-a d) (a+b x)^{m+1} (c+d x)^{-m-1} \left (-a^2 d^2 f^2 \left (m^2+3 m+2\right )+2 a b d f (m+1) (3 d e-c f (1-m))-\left (b^2 \left (c^2 f^2 \left (m^2-3 m+2\right )-6 c d e f (1-m)+6 d^2 e^2\right )\right )\right ) \operatorname {Hypergeometric2F1}\left (2,m+1,m+2,\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{6 (m+1) (b e-a f)^4 (d e-c f)^2}-\frac {f (a+b x)^{m+1} (c+d x)^{1-m} (-a d f (m+2)-b c f (2-m)+4 b d e)}{6 (e+f x)^2 (b e-a f)^2 (d e-c f)^2}-\frac {f (a+b x)^{m+1} (c+d x)^{1-m}}{3 (e+f x)^3 (b e-a f) (d e-c f)} \]
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Rule 12
Rule 105
Rule 133
Rule 156
Rubi steps \begin{align*} \text {integral}& = -\frac {f (a+b x)^{1+m} (c+d x)^{1-m}}{3 (b e-a f) (d e-c f) (e+f x)^3}-\frac {\int \frac {(a+b x)^m (c+d x)^{-m} (-b (3 d e-c f (2-m))+a d f (2+m)+b d f x)}{(e+f x)^3} \, dx}{3 (b e-a f) (d e-c f)} \\ & = -\frac {f (a+b x)^{1+m} (c+d x)^{1-m}}{3 (b e-a f) (d e-c f) (e+f x)^3}-\frac {f (4 b d e-b c f (2-m)-a d f (2+m)) (a+b x)^{1+m} (c+d x)^{1-m}}{6 (b e-a f)^2 (d e-c f)^2 (e+f x)^2}+\frac {\int \frac {\left (-2 a b d f (3 d e-c f (1-m)) (1+m)+a^2 d^2 f^2 \left (2+3 m+m^2\right )+b^2 \left (6 d^2 e^2-6 c d e f (1-m)+c^2 f^2 \left (2-3 m+m^2\right )\right )\right ) (a+b x)^m (c+d x)^{-m}}{(e+f x)^2} \, dx}{6 (b e-a f)^2 (d e-c f)^2} \\ & = -\frac {f (a+b x)^{1+m} (c+d x)^{1-m}}{3 (b e-a f) (d e-c f) (e+f x)^3}-\frac {f (4 b d e-b c f (2-m)-a d f (2+m)) (a+b x)^{1+m} (c+d x)^{1-m}}{6 (b e-a f)^2 (d e-c f)^2 (e+f x)^2}-\frac {\left (2 a b d f (3 d e-c f (1-m)) (1+m)-a^2 d^2 f^2 \left (2+3 m+m^2\right )-b^2 \left (6 d^2 e^2-6 c d e f (1-m)+c^2 f^2 \left (2-3 m+m^2\right )\right )\right ) \int \frac {(a+b x)^m (c+d x)^{-m}}{(e+f x)^2} \, dx}{6 (b e-a f)^2 (d e-c f)^2} \\ & = -\frac {f (a+b x)^{1+m} (c+d x)^{1-m}}{3 (b e-a f) (d e-c f) (e+f x)^3}-\frac {f (4 b d e-b c f (2-m)-a d f (2+m)) (a+b x)^{1+m} (c+d x)^{1-m}}{6 (b e-a f)^2 (d e-c f)^2 (e+f x)^2}-\frac {(b c-a d) \left (2 a b d f (3 d e-c f (1-m)) (1+m)-a^2 d^2 f^2 \left (2+3 m+m^2\right )-b^2 \left (6 d^2 e^2-6 c d e f (1-m)+c^2 f^2 \left (2-3 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-1-m} \, _2F_1\left (2,1+m;2+m;\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{6 (b e-a f)^4 (d e-c f)^2 (1+m)} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b x)^m (c+d x)^{-m}}{(e+f x)^4} \, dx=\frac {(a+b x)^{1+m} (c+d x)^{-1-m} \left (-\frac {2 f (-b e+a f) (-d e+c f) (c+d x)^2}{(e+f x)^3}-\frac {f (4 b d e+b c f (-2+m)-a d f (2+m)) (c+d x)^2}{(e+f x)^2}+\frac {(b c-a d) \left (-2 a b d f (3 d e+c f (-1+m)) (1+m)+a^2 d^2 f^2 \left (2+3 m+m^2\right )+b^2 \left (6 d^2 e^2+6 c d e f (-1+m)+c^2 f^2 \left (2-3 m+m^2\right )\right )\right ) \operatorname {Hypergeometric2F1}\left (2,1+m,2+m,\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(b e-a f)^2 (1+m)}\right )}{6 (b e-a f)^2 (d e-c f)^2} \]
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\[\int \frac {\left (b x +a \right )^{m} \left (d x +c \right )^{-m}}{\left (f x +e \right )^{4}}d x\]
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\[ \int \frac {(a+b x)^m (c+d x)^{-m}}{(e+f x)^4} \, dx=\int { \frac {{\left (b x + a\right )}^{m}}{{\left (f x + e\right )}^{4} {\left (d x + c\right )}^{m}} \,d x } \]
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Timed out. \[ \int \frac {(a+b x)^m (c+d x)^{-m}}{(e+f x)^4} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+b x)^m (c+d x)^{-m}}{(e+f x)^4} \, dx=\int { \frac {{\left (b x + a\right )}^{m}}{{\left (f x + e\right )}^{4} {\left (d x + c\right )}^{m}} \,d x } \]
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\[ \int \frac {(a+b x)^m (c+d x)^{-m}}{(e+f x)^4} \, dx=\int { \frac {{\left (b x + a\right )}^{m}}{{\left (f x + e\right )}^{4} {\left (d x + c\right )}^{m}} \,d x } \]
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Timed out. \[ \int \frac {(a+b x)^m (c+d x)^{-m}}{(e+f x)^4} \, dx=\int \frac {{\left (a+b\,x\right )}^m}{{\left (e+f\,x\right )}^4\,{\left (c+d\,x\right )}^m} \,d x \]
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