Optimal. Leaf size=36 \[ \frac{(d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}{2 e} \]
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Rubi [A] time = 0.0061876, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {609} \[ \frac{(d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}{2 e} \]
Antiderivative was successfully verified.
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Rule 609
Rubi steps
\begin{align*} \int \sqrt{c d^2+2 c d e x+c e^2 x^2} \, dx &=\frac{(d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}{2 e}\\ \end{align*}
Mathematica [A] time = 0.0133653, size = 31, normalized size = 0.86 \[ \frac{c x (d+e x) (2 d+e x)}{2 \sqrt{c (d+e x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 40, normalized size = 1.1 \begin{align*}{\frac{x \left ( ex+2\,d \right ) }{2\,ex+2\,d}\sqrt{c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.02898, size = 90, normalized size = 2.5 \begin{align*} \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}{\left (e x^{2} + 2 \, d x\right )}}{2 \,{\left (e x + d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26372, size = 41, normalized size = 1.14 \begin{align*} \frac{1}{2} \, \sqrt{c x^{2} e^{2} + 2 \, c d x e + c d^{2}}{\left (d e^{\left (-1\right )} + x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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