Optimal. Leaf size=22 \[ \frac {2}{3 \sqrt {3} e \sqrt {2-e x}} \]
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Rubi [A]
time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {641, 32}
\begin {gather*} \frac {2}{3 \sqrt {3} e \sqrt {2-e x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 641
Rubi steps
\begin {align*} \int \frac {(2+e x)^{3/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx &=\int \frac {1}{(6-3 e x)^{3/2}} \, dx\\ &=\frac {2}{3 \sqrt {3} e \sqrt {2-e x}}\\ \end {align*}
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Mathematica [A]
time = 0.23, size = 38, normalized size = 1.73 \begin {gather*} -\frac {2 \sqrt {4-e^2 x^2}}{3 e (-2+e x) \sqrt {6+3 e x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.48, size = 32, normalized size = 1.45
method | result | size |
gosper | \(-\frac {2 \left (e x -2\right ) \left (e x +2\right )^{\frac {3}{2}}}{e \left (-3 e^{2} x^{2}+12\right )^{\frac {3}{2}}}\) | \(30\) |
default | \(-\frac {2 \sqrt {-3 e^{2} x^{2}+12}}{9 \sqrt {e x +2}\, \left (e x -2\right ) e}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.49, size = 15, normalized size = 0.68 \begin {gather*} -\frac {2 i \, \sqrt {3} e^{\left (-1\right )}}{9 \, \sqrt {x e - 2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 34 vs.
\(2 (16) = 32\).
time = 3.36, size = 34, normalized size = 1.55 \begin {gather*} -\frac {2 \, \sqrt {-3 \, x^{2} e^{2} + 12} \sqrt {x e + 2}}{9 \, {\left (x^{2} e^{3} - 4 \, e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\sqrt {3} \left (\int \frac {2 \sqrt {e x + 2}}{- e^{2} x^{2} \sqrt {- e^{2} x^{2} + 4} + 4 \sqrt {- e^{2} x^{2} + 4}}\, dx + \int \frac {e x \sqrt {e x + 2}}{- e^{2} x^{2} \sqrt {- e^{2} x^{2} + 4} + 4 \sqrt {- e^{2} x^{2} + 4}}\, dx\right )}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.97, size = 24, normalized size = 1.09 \begin {gather*} -\frac {1}{9} \, \sqrt {3} e^{\left (-1\right )} + \frac {2 \, \sqrt {3} e^{\left (-1\right )}}{9 \, \sqrt {-x e + 2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.21, size = 24, normalized size = 1.09 \begin {gather*} \frac {2\,\sqrt {e\,x+2}}{3\,e\,\sqrt {12-3\,e^2\,x^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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