Integrand size = 25, antiderivative size = 157 \[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{(f+g x)^2} \, dx=\frac {c (d+e x)^{1+m}}{e g^2 (1+m)}+\frac {\left (a+\frac {f (c f-b g)}{g^2}\right ) (d+e x)^{1+m}}{(e f-d g) (f+g x)}+\frac {(c f (2 d g-e f (2+m))-g (a e g m+b (d g-e f (1+m)))) (d+e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {g (d+e x)}{e f-d g}\right )}{g^2 (e f-d g)^2 (1+m)} \]
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Time = 0.12 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {963, 81, 70} \[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{(f+g x)^2} \, dx=-\frac {(d+e x)^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,-\frac {g (d+e x)}{e f-d g}\right ) (g (a e g m+b d g-b e f (m+1))-c f (2 d g-e f (m+2)))}{g^2 (m+1) (e f-d g)^2}+\frac {(d+e x)^{m+1} \left (a+\frac {f (c f-b g)}{g^2}\right )}{(f+g x) (e f-d g)}+\frac {c (d+e x)^{m+1}}{e g^2 (m+1)} \]
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Rule 70
Rule 81
Rule 963
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a+\frac {f (c f-b g)}{g^2}\right ) (d+e x)^{1+m}}{(e f-d g) (f+g x)}+\frac {\int \frac {(d+e x)^m \left (\frac {c d f g-a e g^2 m-c e f^2 (1+m)-b g (d g-e f (1+m))}{g^2}-c \left (d-\frac {e f}{g}\right ) x\right )}{f+g x} \, dx}{e f-d g} \\ & = \frac {c (d+e x)^{1+m}}{e g^2 (1+m)}+\frac {\left (a+\frac {f (c f-b g)}{g^2}\right ) (d+e x)^{1+m}}{(e f-d g) (f+g x)}-\frac {(g (b d g+a e g m-b e f (1+m))-c f (2 d g-e f (2+m))) \int \frac {(d+e x)^m}{f+g x} \, dx}{g^2 (e f-d g)} \\ & = \frac {c (d+e x)^{1+m}}{e g^2 (1+m)}+\frac {\left (a+\frac {f (c f-b g)}{g^2}\right ) (d+e x)^{1+m}}{(e f-d g) (f+g x)}-\frac {(g (b d g+a e g m-b e f (1+m))-c f (2 d g-e f (2+m))) (d+e x)^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {g (d+e x)}{e f-d g}\right )}{g^2 (e f-d g)^2 (1+m)} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.85 \[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{(f+g x)^2} \, dx=\frac {(d+e x)^{1+m} \left (c (e f-d g)^2-e (2 c f-b g) (e f-d g) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {g (d+e x)}{-e f+d g}\right )+e^2 \left (c f^2+g (-b f+a g)\right ) \operatorname {Hypergeometric2F1}\left (2,1+m,2+m,\frac {g (d+e x)}{-e f+d g}\right )\right )}{e g^2 (e f-d g)^2 (1+m)} \]
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\[\int \frac {\left (e x +d \right )^{m} \left (c \,x^{2}+b x +a \right )}{\left (g x +f \right )^{2}}d x\]
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\[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{(f+g x)^2} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )} {\left (e x + d\right )}^{m}}{{\left (g x + f\right )}^{2}} \,d x } \]
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Exception generated. \[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{(f+g x)^2} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
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\[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{(f+g x)^2} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )} {\left (e x + d\right )}^{m}}{{\left (g x + f\right )}^{2}} \,d x } \]
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\[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{(f+g x)^2} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )} {\left (e x + d\right )}^{m}}{{\left (g x + f\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{(f+g x)^2} \, dx=\int \frac {{\left (d+e\,x\right )}^m\,\left (c\,x^2+b\,x+a\right )}{{\left (f+g\,x\right )}^2} \,d x \]
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