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 = {627, 50, 63, 206} \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 50
Rule 63
Rule 206
Rule 627
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 \operatorname {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 = 0.06, size = 63, normalized size = 1.37 \begin {gather*} \frac {2 \sqrt {12-3 e^2 x^2} \left (\sqrt {e x-2}-2 \tan ^{-1}\left (\frac {1}{2} \sqrt {e x-2}\right )\right )}{e \sqrt {e x-2} \sqrt {e x+2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.28, size = 84, normalized size = 1.83 \begin {gather*} \frac {2 \sqrt {3} \sqrt {4 (e x+2)-(e x+2)^2}}{e \sqrt {e x+2}}-\frac {4 \sqrt {3} \tanh ^{-1}\left (\frac {2 \sqrt {e x+2}}{\sqrt {4 (e x+2)-(e x+2)^2}}\right )}{e} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 99, normalized size = 2.15 \begin {gather*} \frac {2 \, {\left (\sqrt {3} {\left (e x + 2\right )} \log \left (-\frac {3 \, e^{2} x^{2} - 12 \, e x + 4 \, \sqrt {3} \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2} - 36}{e^{2} x^{2} + 4 \, e x + 4}\right ) + \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2}\right )}}{e^{2} x + 2 \, e} \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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maple [A] time = 0.07, size = 66, normalized size = 1.43 \begin {gather*} -\frac {2 \sqrt {-e^{2} x^{2}+4}\, \left (2 \sqrt {3}\, \arctanh \left (\frac {\sqrt {3}\, \sqrt {-3 e x +6}}{6}\right )-\sqrt {-3 e x +6}\right ) \sqrt {3}}{\sqrt {e x +2}\, \sqrt {-3 e x +6}\, e} \end {gather*}
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*} \int \frac {\sqrt {-3 \, e^{2} x^{2} + 12}}{{\left (e x + 2\right )}^{\frac {3}{2}}}\,{d x} \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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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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