Integrand size = 44, antiderivative size = 105 \[ \int \frac {(d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m}}{(f+g x)^3} \, dx=\frac {c^2 d^2 (a e+c d x) (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \operatorname {Hypergeometric2F1}\left (3,1-m,2-m,-\frac {g (a e+c d x)}{c d f-a e g}\right )}{(c d f-a e g)^3 (1-m)} \]
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Time = 0.04 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {905, 70} \[ \int \frac {(d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m}}{(f+g x)^3} \, dx=\frac {c^2 d^2 (d+e x)^m (a e+c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{-m} \operatorname {Hypergeometric2F1}\left (3,1-m,2-m,-\frac {g (a e+c d x)}{c d f-a e g}\right )}{(1-m) (c d f-a e g)^3} \]
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Rule 70
Rule 905
Rubi steps \begin{align*} \text {integral}& = \left ((a e+c d x)^m (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m}\right ) \int \frac {(a e+c d x)^{-m}}{(f+g x)^3} \, dx \\ & = \frac {c^2 d^2 (a e+c d x) (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, _2F_1\left (3,1-m;2-m;-\frac {g (a e+c d x)}{c d f-a e g}\right )}{(c d f-a e g)^3 (1-m)} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.84 \[ \int \frac {(d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m}}{(f+g x)^3} \, dx=-\frac {c^2 d^2 (d+e x)^{-1+m} ((a e+c d x) (d+e x))^{1-m} \operatorname {Hypergeometric2F1}\left (3,1-m,2-m,\frac {g (a e+c d x)}{-c d f+a e g}\right )}{(c d f-a e g)^3 (-1+m)} \]
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\[\int \frac {\left (e x +d \right )^{m} {\left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )}^{-m}}{\left (g x +f \right )^{3}}d x\]
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\[ \int \frac {(d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m}}{(f+g x)^3} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{{\left (g x + f\right )}^{3} {\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{m}} \,d x } \]
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Exception generated. \[ \int \frac {(d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m}}{(f+g x)^3} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
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\[ \int \frac {(d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m}}{(f+g x)^3} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{{\left (g x + f\right )}^{3} {\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{m}} \,d x } \]
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\[ \int \frac {(d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m}}{(f+g x)^3} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{{\left (g x + f\right )}^{3} {\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{m}} \,d x } \]
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Timed out. \[ \int \frac {(d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m}}{(f+g x)^3} \, dx=\int \frac {{\left (d+e\,x\right )}^m}{{\left (f+g\,x\right )}^3\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^m} \,d x \]
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