Integrand size = 35, antiderivative size = 93 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p}{d+e x} \, dx=\frac {\left (-\frac {e (a e+c d x)}{c d^2-a e^2}\right )^{-p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \operatorname {Hypergeometric2F1}\left (-p,p,1+p,\frac {c d (d+e x)}{c d^2-a e^2}\right )}{e p} \]
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Time = 0.05 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {691, 72, 71} \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p}{d+e x} \, dx=\frac {\left (-\frac {e (a e+c d x)}{c d^2-a e^2}\right )^{-p} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^p \operatorname {Hypergeometric2F1}\left (-p,p,p+1,\frac {c d (d+e x)}{c d^2-a e^2}\right )}{e p} \]
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Rule 71
Rule 72
Rule 691
Rubi steps \begin{align*} \text {integral}& = \frac {\left ((a e+c d x)^{-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 (a e+c d x)^p \left (1+\frac {e x}{d}\right )^{-1+p} \, dx}{d} \\ & = \frac {\left (\left (\frac {e (a e+c d x)}{d \left (-c d+\frac {a e^2}{d}\right )}\right )^{-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 (1+\frac {e x}{d}\right )^{-1+p} \left (-\frac {a e^2}{c d^2-a e^2}-\frac {c e x}{c d-\frac {a e^2}{d}}\right )^p \, dx}{d} \\ & = \frac {\left (-\frac {e (a e+c d 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,p;1+p;\frac {c d (d+e x)}{c d^2-a e^2}\right )}{e p} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p}{d+e x} \, dx=\frac {\left (\frac {e (a e+c d x)}{-c d^2+a e^2}\right )^{-p} ((a e+c d x) (d+e x))^p \operatorname {Hypergeometric2F1}\left (-p,p,1+p,\frac {c d (d+e x)}{c d^2-a e^2}\right )}{e p} \]
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\[\int \frac {{\left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )}^{p}}{e x +d}d x\]
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\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p}{d+e x} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{p}}{e x + d} \,d x } \]
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\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p}{d+e x} \, dx=\int \frac {\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{p}}{d + e x}\, dx \]
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\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p}{d+e x} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{p}}{e x + d} \,d x } \]
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\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p}{d+e x} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{p}}{e x + d} \,d x } \]
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Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p}{d+e x} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^p}{d+e\,x} \,d x \]
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