Optimal. Leaf size=38 \[ \frac{(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^p}{e (2 p+1)} \]
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Rubi [A] time = 0.0097498, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {609} \[ \frac{(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^p}{e (2 p+1)} \]
Antiderivative was successfully verified.
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Rule 609
Rubi steps
\begin{align*} \int \left (c d^2+2 c d e x+c e^2 x^2\right )^p \, dx &=\frac{(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^p}{e (1+2 p)}\\ \end{align*}
Mathematica [A] time = 0.0101832, size = 25, normalized size = 0.66 \[ \frac{(d+e x) \left (c (d+e x)^2\right )^p}{2 e p+e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 39, normalized size = 1. \begin{align*}{\frac{ \left ( ex+d \right ) \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{p}}{e \left ( 1+2\,p \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.24028, size = 43, normalized size = 1.13 \begin{align*} \frac{{\left (c^{p} e x + c^{p} d\right )}{\left (e x + d\right )}^{2 \, p}}{e{\left (2 \, p + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.52531, size = 77, normalized size = 2.03 \begin{align*} \frac{{\left (e x + d\right )}{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{p}}{2 \, e p + e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.32154, size = 84, normalized size = 2.21 \begin{align*} \frac{{\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{p} x e +{\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{p} d}{2 \, p e + e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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