Optimal. Leaf size=52 \[ \frac{(d+e x)^{-p-1} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{c d (p+1)} \]
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Rubi [A] time = 0.0160988, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.027, Rules used = {648} \[ \frac{(d+e x)^{-p-1} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{c d (p+1)} \]
Antiderivative was successfully verified.
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Rule 648
Rubi steps
\begin{align*} \int (d+e x)^{-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-p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1+p}}{c d (1+p)}\\ \end{align*}
Mathematica [A] time = 0.0206808, size = 41, normalized size = 0.79 \[ \frac{(d+e x)^{-p-1} ((d+e x) (a e+c d x))^{p+1}}{c d (p+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 56, normalized size = 1.1 \begin{align*}{\frac{ \left ( cdx+ae \right ) \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{p}}{cd \left ( 1+p \right ) \left ( ex+d \right ) ^{p}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02852, size = 41, normalized size = 0.79 \begin{align*} \frac{{\left (c d x + a e\right )}{\left (c d x + a e\right )}^{p}}{c d{\left (p + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.13223, size = 115, normalized size = 2.21 \begin{align*} \frac{{\left (c d x + a e\right )}{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{p}}{{\left (c d p + c d\right )}{\left (e x + d\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 [A] time = 1.14888, size = 115, normalized size = 2.21 \begin{align*} \frac{\frac{c d x e^{\left (p \log \left (c d x + a e\right ) + p \log \left (x e + d\right )\right )}}{{\left (x e + d\right )}^{p}} + \frac{a e^{\left (p \log \left (c d x + a e\right ) + p \log \left (x e + d\right ) + 1\right )}}{{\left (x e + d\right )}^{p}}}{c d p + c d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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