Integrand size = 25, antiderivative size = 245 \[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{(f+g x)^3} \, dx=\frac {\left (a+\frac {f (c f-b g)}{g^2}\right ) (d+e x)^{1+m}}{2 (e f-d g) (f+g x)^2}+\frac {(c f (4 d g-e f (3+m))+g (a e g (1-m)-b (2 d g-e f (1+m)))) (d+e x)^{1+m}}{2 g^2 (e f-d g)^2 (f+g x)}+\frac {\left (c \left (2 d^2 g^2-4 d e f g (1+m)+e^2 f^2 \left (2+3 m+m^2\right )\right )-e g m (a e g (1-m)-b (2 d g-e f (1+m)))\right ) (d+e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {g (d+e x)}{e f-d g}\right )}{2 g^2 (e f-d g)^3 (1+m)} \]
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Time = 0.20 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {963, 79, 70} \[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{(f+g x)^3} \, dx=\frac {(d+e x)^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,-\frac {g (d+e x)}{e f-d g}\right ) \left (e g m (-a e g (1-m)+2 b d g-b e f (m+1))+c \left (2 d^2 g^2-4 d e f g (m+1)+e^2 f^2 \left (m^2+3 m+2\right )\right )\right )}{2 g^2 (m+1) (e f-d g)^3}-\frac {(d+e x)^{m+1} (g (-a e g (1-m)+2 b d g-b e f (m+1))-c f (4 d g-e f (m+3)))}{2 g^2 (f+g x) (e f-d g)^2}+\frac {(d+e x)^{m+1} \left (a+\frac {f (c f-b g)}{g^2}\right )}{2 (f+g x)^2 (e f-d g)} \]
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Rule 70
Rule 79
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}}{2 (e f-d g) (f+g x)^2}+\frac {\int \frac {(d+e x)^m \left (\frac {c f (2 d g-e f (1+m))-g (2 b d g-a e g (1-m)-b e f (1+m))}{g^2}-2 c \left (d-\frac {e f}{g}\right ) x\right )}{(f+g x)^2} \, dx}{2 (e f-d g)} \\ & = \frac {\left (a+\frac {f (c f-b g)}{g^2}\right ) (d+e x)^{1+m}}{2 (e f-d g) (f+g x)^2}-\frac {(g (2 b d g-a e g (1-m)-b e f (1+m))-c f (4 d g-e f (3+m))) (d+e x)^{1+m}}{2 g^2 (e f-d g)^2 (f+g x)}+\frac {\left (e g m (2 b d g-a e g (1-m)-b e f (1+m))+c \left (2 d^2 g^2-4 d e f g (1+m)+e^2 f^2 \left (2+3 m+m^2\right )\right )\right ) \int \frac {(d+e x)^m}{f+g x} \, dx}{2 g^2 (e f-d g)^2} \\ & = \frac {\left (a+\frac {f (c f-b g)}{g^2}\right ) (d+e x)^{1+m}}{2 (e f-d g) (f+g x)^2}-\frac {(g (2 b d g-a e g (1-m)-b e f (1+m))-c f (4 d g-e f (3+m))) (d+e x)^{1+m}}{2 g^2 (e f-d g)^2 (f+g x)}+\frac {\left (e g m (2 b d g-a e g (1-m)-b e f (1+m))+c \left (2 d^2 g^2-4 d e f g (1+m)+e^2 f^2 \left (2+3 m+m^2\right )\right )\right ) (d+e x)^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {g (d+e x)}{e f-d g}\right )}{2 g^2 (e f-d g)^3 (1+m)} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.64 \[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{(f+g x)^3} \, dx=-\frac {(d+e x)^{1+m} \left (c (e f-d g)^2 \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {g (d+e x)}{-e f+d g}\right )+e \left (-\left ((2 c f-b g) (e f-d g) \operatorname {Hypergeometric2F1}\left (2,1+m,2+m,\frac {g (d+e x)}{-e f+d g}\right )\right )+e \left (c f^2+g (-b f+a g)\right ) \operatorname {Hypergeometric2F1}\left (3,1+m,2+m,\frac {g (d+e x)}{-e f+d g}\right )\right )\right )}{g^2 (-e f+d g)^3 (1+m)} \]
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\[\int \frac {\left (e x +d \right )^{m} \left (c \,x^{2}+b x +a \right )}{\left (g x +f \right )^{3}}d x\]
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\[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{(f+g x)^3} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )} {\left (e x + d\right )}^{m}}{{\left (g x + f\right )}^{3}} \,d x } \]
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\[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{(f+g x)^3} \, dx=\int \frac {\left (d + e x\right )^{m} \left (a + b x + c x^{2}\right )}{\left (f + g x\right )^{3}}\, dx \]
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\[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{(f+g x)^3} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )} {\left (e x + d\right )}^{m}}{{\left (g x + f\right )}^{3}} \,d x } \]
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\[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{(f+g x)^3} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )} {\left (e x + d\right )}^{m}}{{\left (g x + f\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{(f+g x)^3} \, dx=\int \frac {{\left (d+e\,x\right )}^m\,\left (c\,x^2+b\,x+a\right )}{{\left (f+g\,x\right )}^3} \,d x \]
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