Integrand size = 37, antiderivative size = 113 \[ \int (d+e x)^{-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx=\frac {(a e+c d x) (d+e x)^{-2 p} \left (\frac {c d (d+e x)}{c d^2-a e^2}\right )^p \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \operatorname {Hypergeometric2F1}\left (p,1+p,2+p,-\frac {e (a e+c d x)}{c d^2-a e^2}\right )}{c d (1+p)} \]
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Time = 0.06 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {693, 691, 72, 71} \[ \int (d+e x)^{-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx=\frac {(d+e x)^{-2 p} (a e+c d x) \left (\frac {c d (d+e x)}{c d^2-a e^2}\right )^p \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^p \operatorname {Hypergeometric2F1}\left (p,p+1,p+2,-\frac {e (a e+c d x)}{c d^2-a e^2}\right )}{c d (p+1)} \]
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Rule 71
Rule 72
Rule 691
Rule 693
Rubi steps \begin{align*} \text {integral}& = \left ((d+e x)^{-2 p} \left (1+\frac {e x}{d}\right )^{2 p}\right ) \int \left (1+\frac {e x}{d}\right )^{-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx \\ & = \left (\left (a d e+c d^2 x\right )^{-p} (d+e x)^{-2 p} \left (1+\frac {e x}{d}\right )^p \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p\right ) \int \left (a d e+c d^2 x\right )^p \left (1+\frac {e x}{d}\right )^{-p} \, dx \\ & = \left (\left (a d e+c d^2 x\right )^{-p} (d+e x)^{-2 p} \left (\frac {c d^2 \left (1+\frac {e x}{d}\right )}{c d^2-a e^2}\right )^p \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p\right ) \int \left (a d e+c d^2 x\right )^p \left (\frac {c d^2}{c d^2-a e^2}+\frac {c d e x}{c d^2-a e^2}\right )^{-p} \, dx \\ & = \frac {(a e+c d x) (d+e x)^{-2 p} \left (\frac {c d (d+e x)}{c d^2-a e^2}\right )^p \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, _2F_1\left (p,1+p;2+p;-\frac {e (a e+c d x)}{c d^2-a e^2}\right )}{c d (1+p)} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.86 \[ \int (d+e x)^{-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx=\frac {(d+e x)^{-1-2 p} \left (\frac {c d (d+e x)}{c d^2-a e^2}\right )^p ((a e+c d x) (d+e x))^{1+p} \operatorname {Hypergeometric2F1}\left (p,1+p,2+p,\frac {e (a e+c d x)}{-c d^2+a e^2}\right )}{c d (1+p)} \]
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\[\int {\left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )}^{p} \left (e x +d \right )^{-2 p}d x\]
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\[ \int (d+e x)^{-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{p}}{{\left (e x + d\right )}^{2 \, p}} \,d x } \]
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Timed out. \[ \int (d+e x)^{-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx=\text {Timed out} \]
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\[ \int (d+e x)^{-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{p}}{{\left (e x + d\right )}^{2 \, p}} \,d x } \]
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\[ \int (d+e x)^{-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{p}}{{\left (e x + d\right )}^{2 \, p}} \,d x } \]
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Timed out. \[ \int (d+e x)^{-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^p}{{\left (d+e\,x\right )}^{2\,p}} \,d x \]
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