Integrand size = 27, antiderivative size = 287 \[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )^2}{f+g x} \, dx=\frac {(b e g-c (e f+d g)) \left (c \left (e^2 f^2+d^2 g^2\right )+e g (2 a e g-b (e f+d g))\right ) (d+e x)^{1+m}}{e^4 g^4 (1+m)}+\frac {\left (b^2 e^2 g^2+c^2 \left (e^2 f^2+2 d e f g+3 d^2 g^2\right )+2 c e g (a e g-b (e f+2 d g))\right ) (d+e x)^{2+m}}{e^4 g^3 (2+m)}-\frac {c (c e f+3 c d g-2 b e g) (d+e x)^{3+m}}{e^4 g^2 (3+m)}+\frac {c^2 (d+e x)^{4+m}}{e^4 g (4+m)}+\frac {\left (c f^2-b f g+a g^2\right )^2 (d+e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {g (d+e x)}{e f-d g}\right )}{g^4 (e f-d g) (1+m)} \]
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Time = 0.54 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {965, 1634, 70} \[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )^2}{f+g x} \, dx=\frac {(d+e x)^{m+2} \left (2 c e g (a e g-b (2 d g+e f))+b^2 e^2 g^2+c^2 \left (3 d^2 g^2+2 d e f g+e^2 f^2\right )\right )}{e^4 g^3 (m+2)}+\frac {(d+e x)^{m+1} (b e g-c (d g+e f)) \left (e g (2 a e g-b (d g+e f))+c \left (d^2 g^2+e^2 f^2\right )\right )}{e^4 g^4 (m+1)}+\frac {(d+e x)^{m+1} \left (a g^2-b f g+c f^2\right )^2 \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,-\frac {g (d+e x)}{e f-d g}\right )}{g^4 (m+1) (e f-d g)}-\frac {c (d+e x)^{m+3} (-2 b e g+3 c d g+c e f)}{e^4 g^2 (m+3)}+\frac {c^2 (d+e x)^{m+4}}{e^4 g (m+4)} \]
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Rule 70
Rule 965
Rule 1634
Rubi steps \begin{align*} \text {integral}& = \frac {c^2 (d+e x)^{4+m}}{e^4 g (4+m)}+\frac {\int \frac {(d+e x)^m \left (-e \left (c^2 d^3 f-a^2 e^3 g\right ) (4+m)+e \left (2 a b e^3 g-c^2 d^2 (3 e f+d g)\right ) (4+m) x+e^2 \left (b^2 e^2 g+2 a c e^2 g-3 c^2 d (e f+d g)\right ) (4+m) x^2-c e^3 (c e f+3 c d g-2 b e g) (4+m) x^3\right )}{f+g x} \, dx}{e^4 g (4+m)} \\ & = \frac {c^2 (d+e x)^{4+m}}{e^4 g (4+m)}+\frac {\int \left (\frac {e (b e g-c (e f+d g)) \left (c \left (e^2 f^2+d^2 g^2\right )+e g (2 a e g-b (e f+d g))\right ) (4+m) (d+e x)^m}{g^3}+\frac {e \left (b^2 e^2 g^2+c^2 \left (e^2 f^2+2 d e f g+3 d^2 g^2\right )+2 c e g (a e g-b (e f+2 d g))\right ) (4+m) (d+e x)^{1+m}}{g^2}-\frac {c e (c e f+3 c d g-2 b e g) (4+m) (d+e x)^{2+m}}{g}+\frac {e^4 \left (c f^2-b f g+a g^2\right )^2 (4+m) (d+e x)^m}{g^3 (f+g x)}\right ) \, dx}{e^4 g (4+m)} \\ & = \frac {(b e g-c (e f+d g)) \left (c \left (e^2 f^2+d^2 g^2\right )+e g (2 a e g-b (e f+d g))\right ) (d+e x)^{1+m}}{e^4 g^4 (1+m)}+\frac {\left (b^2 e^2 g^2+c^2 \left (e^2 f^2+2 d e f g+3 d^2 g^2\right )+2 c e g (a e g-b (e f+2 d g))\right ) (d+e x)^{2+m}}{e^4 g^3 (2+m)}-\frac {c (c e f+3 c d g-2 b e g) (d+e x)^{3+m}}{e^4 g^2 (3+m)}+\frac {c^2 (d+e x)^{4+m}}{e^4 g (4+m)}+\frac {\left (c f^2-b f g+a g^2\right )^2 \int \frac {(d+e x)^m}{f+g x} \, dx}{g^4} \\ & = \frac {(b e g-c (e f+d g)) \left (c \left (e^2 f^2+d^2 g^2\right )+e g (2 a e g-b (e f+d g))\right ) (d+e x)^{1+m}}{e^4 g^4 (1+m)}+\frac {\left (b^2 e^2 g^2+c^2 \left (e^2 f^2+2 d e f g+3 d^2 g^2\right )+2 c e g (a e g-b (e f+2 d g))\right ) (d+e x)^{2+m}}{e^4 g^3 (2+m)}-\frac {c (c e f+3 c d g-2 b e g) (d+e x)^{3+m}}{e^4 g^2 (3+m)}+\frac {c^2 (d+e x)^{4+m}}{e^4 g (4+m)}+\frac {\left (c f^2-b f g+a g^2\right )^2 (d+e x)^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {g (d+e x)}{e f-d g}\right )}{g^4 (e f-d g) (1+m)} \\ \end{align*}
Time = 0.56 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.92 \[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )^2}{f+g x} \, dx=\frac {(d+e x)^{1+m} \left (-\frac {(c e f+c d g-b e g) \left (c \left (e^2 f^2+d^2 g^2\right )+e g (2 a e g-b (e f+d g))\right )}{e^4 (1+m)}+\frac {g \left (b^2 e^2 g^2+c^2 \left (e^2 f^2+2 d e f g+3 d^2 g^2\right )+2 c e g (a e g-b (e f+2 d g))\right ) (d+e x)}{e^4 (2+m)}-\frac {c g^2 (c e f+3 c d g-2 b e g) (d+e x)^2}{e^4 (3+m)}+\frac {c^2 g^3 (d+e x)^3}{e^4 (4+m)}+\frac {\left (c f^2+g (-b f+a g)\right )^2 \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {g (d+e x)}{-e f+d g}\right )}{(e f-d g) (1+m)}\right )}{g^4} \]
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\[\int \frac {\left (e x +d \right )^{m} \left (c \,x^{2}+b x +a \right )^{2}}{g x +f}d x\]
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\[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )^2}{f+g x} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{2} {\left (e x + d\right )}^{m}}{g x + f} \,d x } \]
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\[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )^2}{f+g x} \, dx=\int \frac {\left (d + e x\right )^{m} \left (a + b x + c x^{2}\right )^{2}}{f + g x}\, dx \]
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\[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )^2}{f+g x} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{2} {\left (e x + d\right )}^{m}}{g x + f} \,d x } \]
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\[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )^2}{f+g x} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{2} {\left (e x + d\right )}^{m}}{g x + f} \,d x } \]
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Timed out. \[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )^2}{f+g x} \, dx=\int \frac {{\left (d+e\,x\right )}^m\,{\left (c\,x^2+b\,x+a\right )}^2}{f+g\,x} \,d x \]
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