Optimal. Leaf size=39 \[ \frac{(d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}{2 c e} \]
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Rubi [A] time = 0.0215158, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {642, 609} \[ \frac{(d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}{2 c e} \]
Antiderivative was successfully verified.
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Rule 642
Rule 609
Rubi steps
\begin{align*} \int \frac{(d+e x)^2}{\sqrt{c d^2+2 c d e x+c e^2 x^2}} \, dx &=\frac{\int \sqrt{c d^2+2 c d e x+c e^2 x^2} \, dx}{c}\\ &=\frac{(d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}{2 c e}\\ \end{align*}
Mathematica [A] time = 0.003141, size = 30, normalized size = 0.77 \[ \frac{x (d+e x) (2 d+e x)}{2 \sqrt{c (d+e x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 38, normalized size = 1. \begin{align*}{\frac{x \left ( ex+2\,d \right ) \left ( ex+d \right ) }{2}{\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 [B] time = 1.16027, size = 147, normalized size = 3.77 \begin{align*} \frac{c^{2} d^{2} e^{4} \log \left (x + \frac{d}{e}\right )}{\left (c e^{2}\right )^{\frac{5}{2}}} - \frac{c d e^{3} x}{\left (c e^{2}\right )^{\frac{3}{2}}} + \frac{e^{2} x^{2}}{2 \, \sqrt{c e^{2}}} - d^{2} \sqrt{\frac{1}{c e^{2}}} \log \left (x + \frac{d}{e}\right ) + \frac{2 \, \sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} d}{c e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.38543, size = 96, normalized size = 2.46 \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 (c e x + c 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 \frac{\left (d + e x\right )^{2}}{\sqrt{c \left (d + e x\right )^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.39398, size = 50, normalized size = 1.28 \begin{align*} \frac{1}{2} \, \sqrt{c x^{2} e^{2} + 2 \, c d x e + c d^{2}}{\left (\frac{d e^{\left (-1\right )}}{c} + \frac{x}{c}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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