Optimal. Leaf size=32 \[ \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^p}{2 e p} \]
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Rubi [A] time = 0.0212296, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {643, 629} \[ \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^p}{2 e p} \]
Antiderivative was successfully verified.
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Rule 643
Rule 629
Rubi steps
\begin{align*} \int \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^p}{d+e x} \, dx &=c \int (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{-1+p} \, dx\\ &=\frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^p}{2 e p}\\ \end{align*}
Mathematica [A] time = 0.0064535, size = 21, normalized size = 0.66 \[ \frac{\left (c (d+e x)^2\right )^p}{2 e p} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 31, normalized size = 1. \begin{align*}{\frac{ \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{p}}{2\,ep}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15541, size = 27, normalized size = 0.84 \begin{align*} \frac{{\left (e x + d\right )}^{2 \, p} c^{p}}{2 \, e p} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.51845, size = 61, normalized size = 1.91 \begin{align*} \frac{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{p}}{2 \, e p} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.348116, size = 48, normalized size = 1.5 \begin{align*} \begin{cases} \frac{x}{d} & \text{for}\: e = 0 \wedge p = 0 \\\frac{\log{\left (\frac{d}{e} + x \right )}}{e} & \text{for}\: p = 0 \\\frac{x \left (c d^{2}\right )^{p}}{d} & \text{for}\: e = 0 \\\frac{\left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{p}}{2 e p} & \text{otherwise} \end{cases} \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 \frac{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{p}}{e x + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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