Optimal. Leaf size=46 \[ \frac {2 \sqrt {3} \sqrt {2-e x}}{e}-\frac {4 \sqrt {3} \tanh ^{-1}\left (\frac {1}{2} \sqrt {2-e x}\right )}{e} \]
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Rubi [A]
time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {641, 52, 65,
212} \begin {gather*} \frac {2 \sqrt {3} \sqrt {2-e x}}{e}-\frac {4 \sqrt {3} \tanh ^{-1}\left (\frac {1}{2} \sqrt {2-e x}\right )}{e} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 212
Rule 641
Rubi steps
\begin {align*} \int \frac {\sqrt {12-3 e^2 x^2}}{(2+e x)^{3/2}} \, dx &=\int \frac {\sqrt {6-3 e x}}{2+e x} \, dx\\ &=\frac {2 \sqrt {3} \sqrt {2-e x}}{e}+12 \int \frac {1}{\sqrt {6-3 e x} (2+e x)} \, dx\\ &=\frac {2 \sqrt {3} \sqrt {2-e x}}{e}-\frac {8 \text {Subst}\left (\int \frac {1}{4-\frac {x^2}{3}} \, dx,x,\sqrt {6-3 e x}\right )}{e}\\ &=\frac {2 \sqrt {3} \sqrt {2-e x}}{e}-\frac {4 \sqrt {3} \tanh ^{-1}\left (\frac {1}{2} \sqrt {2-e x}\right )}{e}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 63, normalized size = 1.37 \begin {gather*} \frac {2 \sqrt {3} \left (\frac {\sqrt {4-e^2 x^2}}{\sqrt {2+e x}}-2 \tanh ^{-1}\left (\frac {2 \sqrt {2+e x}}{\sqrt {4-e^2 x^2}}\right )\right )}{e} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.58, size = 66, normalized size = 1.43
method | result | size |
default | \(-\frac {2 \sqrt {-e^{2} x^{2}+4}\, \left (2 \sqrt {3}\, \arctanh \left (\frac {\sqrt {-3 e x +6}\, \sqrt {3}}{6}\right )-\sqrt {-3 e x +6}\right ) \sqrt {3}}{\sqrt {e x +2}\, \sqrt {-3 e x +6}\, e}\) | \(66\) |
risch | \(-\frac {6 \left (e x -2\right ) \sqrt {\frac {-3 e^{2} x^{2}+12}{e x +2}}\, \sqrt {e x +2}}{e \sqrt {-3 e x +6}\, \sqrt {-3 e^{2} x^{2}+12}}-\frac {4 \sqrt {3}\, \arctanh \left (\frac {\sqrt {-3 e x +6}\, \sqrt {3}}{6}\right ) \sqrt {\frac {-3 e^{2} x^{2}+12}{e x +2}}\, \sqrt {e x +2}}{e \sqrt {-3 e^{2} x^{2}+12}}\) | \(120\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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. 100 vs.
\(2 (36) = 72\).
time = 2.20, size = 100, normalized size = 2.17 \begin {gather*} \frac {2 \, {\left (\sqrt {3} {\left (x e + 2\right )} \log \left (-\frac {3 \, x^{2} e^{2} - 12 \, x e + 4 \, \sqrt {3} \sqrt {-3 \, x^{2} e^{2} + 12} \sqrt {x e + 2} - 36}{x^{2} e^{2} + 4 \, x e + 4}\right ) + \sqrt {-3 \, x^{2} e^{2} + 12} \sqrt {x e + 2}\right )}}{x e^{2} + 2 \, e} \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*} \sqrt {3} \int \frac {\sqrt {- e^{2} x^{2} + 4}}{e x \sqrt {e x + 2} + 2 \sqrt {e x + 2}}\, dx \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 [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {12-3\,e^2\,x^2}}{{\left (e\,x+2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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