Optimal. Leaf size=50 \[ \frac {1}{6 \sqrt {3} e \sqrt {2-e x}}-\frac {\tanh ^{-1}\left (\frac {1}{2} \sqrt {2-e x}\right )}{12 \sqrt {3} e} \]
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Rubi [A] time = 0.02, antiderivative size = 50, 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, 51, 63, 206} \begin {gather*} \frac {1}{6 \sqrt {3} e \sqrt {2-e x}}-\frac {\tanh ^{-1}\left (\frac {1}{2} \sqrt {2-e x}\right )}{12 \sqrt {3} e} \end {gather*}
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 206
Rule 627
Rubi steps
\begin {align*} \int \frac {\sqrt {2+e x}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx &=\int \frac {1}{(6-3 e x)^{3/2} (2+e x)} \, dx\\ &=\frac {1}{6 \sqrt {3} e \sqrt {2-e x}}+\frac {1}{12} \int \frac {1}{\sqrt {6-3 e x} (2+e x)} \, dx\\ &=\frac {1}{6 \sqrt {3} e \sqrt {2-e x}}-\frac {\operatorname {Subst}\left (\int \frac {1}{4-\frac {x^2}{3}} \, dx,x,\sqrt {6-3 e x}\right )}{18 e}\\ &=\frac {1}{6 \sqrt {3} e \sqrt {2-e x}}-\frac {\tanh ^{-1}\left (\frac {1}{2} \sqrt {2-e x}\right )}{12 \sqrt {3} e}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 48, normalized size = 0.96 \begin {gather*} \frac {\sqrt {e x+2} \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {1}{2}-\frac {e x}{4}\right )}{6 e \sqrt {12-3 e^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.34, size = 95, normalized size = 1.90 \begin {gather*} -\frac {\sqrt {4 (e x+2)-(e x+2)^2}}{6 \sqrt {3} e (e x-2) \sqrt {e x+2}}-\frac {\tanh ^{-1}\left (\frac {2 \sqrt {e x+2}}{\sqrt {4 (e x+2)-(e x+2)^2}}\right )}{12 \sqrt {3} e} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 106, normalized size = 2.12 \begin {gather*} \frac {\sqrt {3} {\left (e^{2} x^{2} - 4\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 ) - 4 \, \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2}}{72 \, {\left (e^{3} x^{2} - 4 \, e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 60, normalized size = 1.20 \begin {gather*} \frac {\sqrt {-3 e^{2} x^{2}+12}\, \left (\sqrt {3}\, \sqrt {-3 e x +6}\, \arctanh \left (\frac {\sqrt {3}\, \sqrt {-3 e x +6}}{6}\right )-6\right )}{108 \sqrt {e x +2}\, \left (e x -2\right ) 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 {e x + 2}}{{\left (-3 \, e^{2} x^{2} + 12\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 {e\,x+2}}{{\left (12-3\,e^2\,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*} \frac {\sqrt {3} \int \frac {\sqrt {e x + 2}}{- e^{2} x^{2} \sqrt {- e^{2} x^{2} + 4} + 4 \sqrt {- e^{2} x^{2} + 4}}\, dx}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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