Optimal. Leaf size=107 \[ -\frac{(d+e x)^{-2 p} \left (-\frac{e (a e+c d x)}{c d^2-a e^2}\right )^{-p} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^p \, _2F_1\left (-p,-p;1-p;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{e p} \]
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Rubi [A] time = 0.095885, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {679, 677, 70, 69} \[ -\frac{(d+e x)^{-2 p} \left (-\frac{e (a e+c d x)}{c d^2-a e^2}\right )^{-p} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^p \, _2F_1\left (-p,-p;1-p;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{e p} \]
Antiderivative was successfully verified.
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Rule 679
Rule 677
Rule 70
Rule 69
Rubi steps
\begin{align*} \int (d+e x)^{-1-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx &=\frac{\left ((d+e x)^{-2 p} \left (1+\frac{e x}{d}\right )^{2 p}\right ) \int \left (1+\frac{e x}{d}\right )^{-1-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx}{d}\\ &=\frac{\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 )^{-1-p} \, dx}{d}\\ &=\frac{\left (\left (\frac{e \left (a d e+c d^2 x\right )}{d \left (-c d^2+a e^2\right )}\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 (1+\frac{e x}{d}\right )^{-1-p} \left (-\frac{a e^2}{c d^2-a e^2}-\frac{c d e x}{c d^2-a e^2}\right )^p \, dx}{d}\\ &=-\frac{\left (-\frac{e (a e+c d x)}{c d^2-a e^2}\right )^{-p} (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 \, _2F_1\left (-p,-p;1-p;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{e p}\\ \end{align*}
Mathematica [A] time = 0.0222295, size = 95, normalized size = 0.89 \[ -\frac{(d+e x)^{-2 p} \left (\frac{e (a e+c d x)}{a e^2-c d^2}\right )^{-p} ((d+e x) (a e+c d x))^p \, _2F_1\left (-p,-p;1-p;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{e p} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.219, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) ^{-1-2\,p} \left ( ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2} \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\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 - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\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 - 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\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 - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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