Optimal. Leaf size=60 \[ \frac{(d+e x)^{-2 (p+1)} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+1) \left (c d^2-a e^2\right )} \]
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Rubi [A] time = 0.0162893, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.026, Rules used = {650} \[ \frac{(d+e x)^{-2 (p+1)} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+1) \left (c d^2-a e^2\right )} \]
Antiderivative was successfully verified.
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Rule 650
Rubi steps
\begin{align*} \int (d+e x)^{-2-2 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 (1+p)} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1+p}}{\left (c d^2-a e^2\right ) (1+p)}\\ \end{align*}
Mathematica [A] time = 0.0286327, size = 49, normalized size = 0.82 \[ \frac{(d+e x)^{-2 (p+1)} ((d+e x) (a e+c d x))^{p+1}}{(p+1) \left (c d^2-a e^2\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 75, normalized size = 1.3 \begin{align*} -{\frac{ \left ( cdx+ae \right ) \left ( ex+d \right ) ^{-1-2\,p} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{p}}{a{e}^{2}p-c{d}^{2}p+a{e}^{2}-c{d}^{2}}} \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 - 2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.22, size = 189, normalized size = 3.15 \begin{align*} \frac{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}{\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 - 2}}{c d^{2} - a e^{2} +{\left (c d^{2} - a e^{2}\right )} p} \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 - 2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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