Optimal. Leaf size=31 \[ \frac{\sqrt{c d^2+2 c d e x+c e^2 x^2}}{c e} \]
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Rubi [A] time = 0.0087882, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.033, Rules used = {629} \[ \frac{\sqrt{c d^2+2 c d e x+c e^2 x^2}}{c e} \]
Antiderivative was successfully verified.
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Rule 629
Rubi steps
\begin{align*} \int \frac{d+e x}{\sqrt{c d^2+2 c d e x+c e^2 x^2}} \, dx &=\frac{\sqrt{c d^2+2 c d e x+c e^2 x^2}}{c e}\\ \end{align*}
Mathematica [A] time = 0.0025526, size = 20, normalized size = 0.65 \[ \frac{x (d+e x)}{\sqrt{c (d+e x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 30, normalized size = 1. \begin{align*}{ \left ( ex+d \right ) x{\frac{1}{\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 [A] time = 1.20373, size = 39, normalized size = 1.26 \begin{align*} \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{c e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.27197, size = 72, normalized size = 2.32 \begin{align*} \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} x}{c e x + c d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.827076, size = 39, normalized size = 1.26 \begin{align*} \begin{cases} \frac{\sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{c e} & \text{for}\: e \neq 0 \\\frac{d x}{\sqrt{c d^{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.34488, size = 38, normalized size = 1.23 \begin{align*} \frac{\sqrt{c x^{2} e^{2} + 2 \, c d x e + c d^{2}} e^{\left (-1\right )}}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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