Optimal. Leaf size=43 \[ -\frac {8 \sqrt {2-e x}}{\sqrt {3} e}+\frac {2 (2-e x)^{3/2}}{3 \sqrt {3} e} \]
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Rubi [A]
time = 0.01, antiderivative size = 43, 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 (2-e x)^{3/2}}{3 \sqrt {3} e}-\frac {8 \sqrt {2-e x}}{\sqrt {3} e} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 641
Rubi steps
\begin {align*} \int \frac {(2+e x)^{3/2}}{\sqrt {12-3 e^2 x^2}} \, dx &=\int \frac {2+e x}{\sqrt {6-3 e x}} \, dx\\ &=\int \left (\frac {4}{\sqrt {6-3 e x}}-\frac {1}{3} \sqrt {6-3 e x}\right ) \, dx\\ &=-\frac {8 \sqrt {2-e x}}{\sqrt {3} e}+\frac {2 (2-e x)^{3/2}}{3 \sqrt {3} e}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 36, normalized size = 0.84 \begin {gather*} -\frac {2 (10+e x) \sqrt {4-e^2 x^2}}{3 e \sqrt {6+3 e x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.46, size = 30, normalized size = 0.70
method | result | size |
default | \(-\frac {2 \sqrt {-3 e^{2} x^{2}+12}\, \left (e x +10\right )}{9 \sqrt {e x +2}\, e}\) | \(30\) |
gosper | \(\frac {2 \left (e x -2\right ) \left (e x +10\right ) \sqrt {e x +2}}{3 e \sqrt {-3 e^{2} x^{2}+12}}\) | \(35\) |
risch | \(\frac {2 \sqrt {\frac {-3 e^{2} x^{2}+12}{e x +2}}\, \sqrt {e x +2}\, \left (e x +10\right ) \left (e x -2\right )}{3 \sqrt {-3 e^{2} x^{2}+12}\, e \sqrt {-3 e x +6}}\) | \(63\) |
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 = 28, normalized size = 0.65 \begin {gather*} -\frac {2 i \, \sqrt {3} {\left (x^{2} e^{2} + 8 \, x e - 20\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 = 2.79, size = 38, normalized size = 0.88 \begin {gather*} -\frac {2 \, \sqrt {-3 \, x^{2} e^{2} + 12} {\left (x e + 10\right )} \sqrt {x e + 2}}{9 \, {\left (x e^{2} + 2 \, 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}}{\sqrt {- e^{2} x^{2} + 4}}\, dx + \int \frac {e x \sqrt {e x + 2}}{\sqrt {- e^{2} x^{2} + 4}}\, dx\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.14, size = 49, normalized size = 1.14 \begin {gather*} -\frac {\left (\frac {20\,\sqrt {e\,x+2}}{9\,e^2}+\frac {2\,x\,\sqrt {e\,x+2}}{9\,e}\right )\,\sqrt {12-3\,e^2\,x^2}}{x+\frac {2}{e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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