Optimal. Leaf size=34 \[ \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{3 c e} \]
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Rubi [A] time = 0.0087348, antiderivative size = 34, 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} \[ \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{3 c e} \]
Antiderivative was successfully verified.
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Rule 629
Rubi steps
\begin{align*} \int (d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2} \, dx &=\frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{3 c e}\\ \end{align*}
Mathematica [A] time = 0.0086244, size = 23, normalized size = 0.68 \[ \frac{\left (c (d+e x)^2\right )^{3/2}}{3 c e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 51, normalized size = 1.5 \begin{align*}{\frac{x \left ({e}^{2}{x}^{2}+3\,dex+3\,{d}^{2} \right ) }{3\,ex+3\,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 [A] time = 0.985503, size = 41, normalized size = 1.21 \begin{align*} \frac{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac{3}{2}}}{3 \, c e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.04189, size = 112, normalized size = 3.29 \begin{align*} \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}{\left (e^{2} x^{3} + 3 \, d e x^{2} + 3 \, d^{2} x\right )}}{3 \,{\left (e x + d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.292331, size = 107, normalized size = 3.15 \begin{align*} \begin{cases} \frac{d^{2} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{3 e} + \frac{2 d x \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{3} + \frac{e x^{2} \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{3} & \text{for}\: e \neq 0 \\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 = 1.16425, size = 55, normalized size = 1.62 \begin{align*} \frac{1}{3} \, \sqrt{c x^{2} e^{2} + 2 \, c d x e + c d^{2}}{\left (d^{2} e^{\left (-1\right )} +{\left (x e + 2 \, d\right )} x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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