Optimal. Leaf size=57 \[ -\frac {\sqrt {2-e x}}{4 \sqrt {3} e (e x+2)}-\frac {\tanh ^{-1}\left (\frac {1}{2} \sqrt {2-e x}\right )}{8 \sqrt {3} e} \]
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Rubi [A] time = 0.02, antiderivative size = 57, 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 {\sqrt {2-e x}}{4 \sqrt {3} e (e x+2)}-\frac {\tanh ^{-1}\left (\frac {1}{2} \sqrt {2-e x}\right )}{8 \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 {1}{(2+e x)^{3/2} \sqrt {12-3 e^2 x^2}} \, dx &=\int \frac {1}{\sqrt {6-3 e x} (2+e x)^2} \, dx\\ &=-\frac {\sqrt {2-e x}}{4 \sqrt {3} e (2+e x)}+\frac {1}{8} \int \frac {1}{\sqrt {6-3 e x} (2+e x)} \, dx\\ &=-\frac {\sqrt {2-e x}}{4 \sqrt {3} e (2+e x)}-\frac {\operatorname {Subst}\left (\int \frac {1}{4-\frac {x^2}{3}} \, dx,x,\sqrt {6-3 e x}\right )}{12 e}\\ &=-\frac {\sqrt {2-e x}}{4 \sqrt {3} e (2+e x)}-\frac {\tanh ^{-1}\left (\frac {1}{2} \sqrt {2-e x}\right )}{8 \sqrt {3} e}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 54, normalized size = 0.95 \begin {gather*} \frac {-2 \sqrt {2-e x}-\left ((e x+2) \tanh ^{-1}\left (\frac {1}{2} \sqrt {2-e x}\right )\right )}{8 \sqrt {3} e (e x+2)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 0.35, size = 164, normalized size = 2.88 \begin {gather*} \frac {\frac {\sqrt {4 (e x+2)-(e x+2)^2}}{4 \sqrt {3} e \sqrt {e x+2}}+\frac {(e x+2) \tanh ^{-1}\left (\frac {2 \sqrt {e x+2}}{\sqrt {4 (e x+2)-(e x+2)^2}}\right )}{8 \sqrt {3} e}}{\left (\frac {\sqrt {4 (e x+2)-(e x+2)^2}}{\sqrt {e x+2}}-2\right ) \left (\frac {\sqrt {4 (e x+2)-(e x+2)^2}}{\sqrt {e x+2}}+2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 116, normalized size = 2.04 \begin {gather*} \frac {\sqrt {3} {\left (e^{2} x^{2} + 4 \, e x + 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}}{48 \, {\left (e^{3} x^{2} + 4 \, e^{2} x + 4 \, e\right )}} \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 [B] time = 0.06, size = 88, normalized size = 1.54 \begin {gather*} -\frac {\sqrt {-e^{2} x^{2}+4}\, \left (\sqrt {3}\, e x \arctanh \left (\frac {\sqrt {3}\, \sqrt {-3 e x +6}}{6}\right )+2 \sqrt {3}\, \arctanh \left (\frac {\sqrt {3}\, \sqrt {-3 e x +6}}{6}\right )+2 \sqrt {-3 e x +6}\right ) \sqrt {3}}{24 \sqrt {\left (e x +2\right )^{3}}\, \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 {1}{\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 {1}{\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*} \frac {\sqrt {3} \int \frac {1}{e x \sqrt {e x + 2} \sqrt {- e^{2} x^{2} + 4} + 2 \sqrt {e x + 2} \sqrt {- e^{2} x^{2} + 4}}\, dx}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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