Integrand size = 25, antiderivative size = 129 \[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{f+g x} \, dx=-\frac {(c e f+c d g-b e g) (d+e x)^{1+m}}{e^2 g^2 (1+m)}+\frac {c (d+e x)^{2+m}}{e^2 g (2+m)}+\frac {\left (c f^2-b f g+a g^2\right ) (d+e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {g (d+e x)}{e f-d g}\right )}{g^2 (e f-d g) (1+m)} \]
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Time = 0.10 (sec) , antiderivative size = 129, 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 = {965, 81, 70} \[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{f+g x} \, dx=\frac {(d+e x)^{m+1} \left (a g^2-b f g+c f^2\right ) \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,-\frac {g (d+e x)}{e f-d g}\right )}{g^2 (m+1) (e f-d g)}-\frac {(d+e x)^{m+1} (-b e g+c d g+c e f)}{e^2 g^2 (m+1)}+\frac {c (d+e x)^{m+2}}{e^2 g (m+2)} \]
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Rule 70
Rule 81
Rule 965
Rubi steps \begin{align*} \text {integral}& = \frac {c (d+e x)^{2+m}}{e^2 g (2+m)}+\frac {\int \frac {(d+e x)^m (-e (c d f-a e g) (2+m)-e (c e f+c d g-b e g) (2+m) x)}{f+g x} \, dx}{e^2 g (2+m)} \\ & = -\frac {(c e f+c d g-b e g) (d+e x)^{1+m}}{e^2 g^2 (1+m)}+\frac {c (d+e x)^{2+m}}{e^2 g (2+m)}+\frac {\left (c f^2-b f g+a g^2\right ) \int \frac {(d+e x)^m}{f+g x} \, dx}{g^2} \\ & = -\frac {(c e f+c d g-b e g) (d+e x)^{1+m}}{e^2 g^2 (1+m)}+\frac {c (d+e x)^{2+m}}{e^2 g (2+m)}+\frac {\left (c f^2-b f g+a g^2\right ) (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) (1+m)} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.86 \[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{f+g x} \, dx=\frac {(d+e x)^{1+m} \left (\frac {b e g-c (e f+d g)}{e^2 (1+m)}+\frac {c g (d+e x)}{e^2 (2+m)}+\frac {\left (c f^2+g (-b f+a g)\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {g (d+e x)}{-e f+d g}\right )}{(e f-d g) (1+m)}\right )}{g^2} \]
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\[\int \frac {\left (e x +d \right )^{m} \left (c \,x^{2}+b x +a \right )}{g x +f}d x\]
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\[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{f+g x} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )} {\left (e x + d\right )}^{m}}{g x + f} \,d x } \]
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\[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{f+g x} \, dx=\int \frac {\left (d + e x\right )^{m} \left (a + b x + c x^{2}\right )}{f + g x}\, dx \]
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\[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{f+g x} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )} {\left (e x + d\right )}^{m}}{g x + f} \,d x } \]
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\[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{f+g x} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )} {\left (e x + d\right )}^{m}}{g x + f} \,d x } \]
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Timed out. \[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{f+g x} \, dx=\int \frac {{\left (d+e\,x\right )}^m\,\left (c\,x^2+b\,x+a\right )}{f+g\,x} \,d x \]
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