Optimal. Leaf size=45 \[ \frac {8}{3 \sqrt {3} e \sqrt {2-e x}}+\frac {2 \sqrt {2-e x}}{3 \sqrt {3} e} \]
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Rubi [A]
time = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {641, 45}
\begin {gather*} \frac {2 \sqrt {2-e x}}{3 \sqrt {3} e}+\frac {8}{3 \sqrt {3} e \sqrt {2-e x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 641
Rubi steps
\begin {align*} \int \frac {(2+e x)^{5/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx &=\int \frac {2+e x}{(6-3 e x)^{3/2}} \, dx\\ &=\int \left (\frac {4}{(6-3 e x)^{3/2}}-\frac {1}{3 \sqrt {6-3 e x}}\right ) \, dx\\ &=\frac {8}{3 \sqrt {3} e \sqrt {2-e x}}+\frac {2 \sqrt {2-e x}}{3 \sqrt {3} e}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 43, normalized size = 0.96 \begin {gather*} \frac {2 (-6+e x) \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 = 0.49, size = 37, normalized size = 0.82
method | result | size |
gosper | \(\frac {2 \left (e x -2\right ) \left (e x -6\right ) \left (e x +2\right )^{\frac {3}{2}}}{e \left (-3 e^{2} x^{2}+12\right )^{\frac {3}{2}}}\) | \(35\) |
default | \(\frac {2 \sqrt {-3 e^{2} x^{2}+12}\, \left (e x -6\right )}{9 \sqrt {e x +2}\, \left (e x -2\right ) e}\) | \(37\) |
risch | \(-\frac {2 \left (e x -2\right ) \sqrt {\frac {-3 e^{2} x^{2}+12}{e x +2}}\, \sqrt {e x +2}}{3 e \sqrt {-3 e x +6}\, \sqrt {-3 e^{2} x^{2}+12}}+\frac {8 \sqrt {\frac {-3 e^{2} x^{2}+12}{e x +2}}\, \sqrt {e x +2}}{3 e \sqrt {-3 e x +6}\, \sqrt {-3 e^{2} x^{2}+12}}\) | \(111\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.53, size = 21, normalized size = 0.47 \begin {gather*} \frac {2 i \, \sqrt {3} {\left (x e - 6\right )} 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 [A]
time = 5.32, size = 40, normalized size = 0.89 \begin {gather*} \frac {2 \, \sqrt {-3 \, x^{2} e^{2} + 12} \sqrt {x e + 2} {\left (x e - 6\right )}}{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 {4 \sqrt {e x + 2}}{- e^{2} x^{2} \sqrt {- e^{2} x^{2} + 4} + 4 \sqrt {- e^{2} x^{2} + 4}}\, dx + \int \frac {4 e x \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^{2} x^{2} \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 = 1.16, size = 40, normalized size = 0.89 \begin {gather*} \frac {2}{9} \, \sqrt {3} \sqrt {-x e + 2} e^{\left (-1\right )} - \frac {8}{9} \, \sqrt {3} e^{\left (-1\right )} + \frac {8 \, \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 = 0.57, size = 52, normalized size = 1.16 \begin {gather*} \frac {\left (\frac {4\,\sqrt {e\,x+2}}{3\,e^3}-\frac {2\,x\,\sqrt {e\,x+2}}{9\,e^2}\right )\,\sqrt {12-3\,e^2\,x^2}}{\frac {4}{e^2}-x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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