Optimal. Leaf size=45 \[ \frac {6 \sqrt {3} (2-e x)^{7/2}}{7 e}-\frac {24 \sqrt {3} (2-e x)^{5/2}}{5 e} \]
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Rubi [A] time = 0.02, antiderivative size = 45, 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 = {627, 43} \[ \frac {6 \sqrt {3} (2-e x)^{7/2}}{7 e}-\frac {24 \sqrt {3} (2-e x)^{5/2}}{5 e} \]
Antiderivative was successfully verified.
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Rule 43
Rule 627
Rubi steps
\begin {align*} \int \frac {\left (12-3 e^2 x^2\right )^{3/2}}{\sqrt {2+e x}} \, dx &=\int (6-3 e x)^{3/2} (2+e x) \, dx\\ &=\int \left (4 (6-3 e x)^{3/2}-\frac {1}{3} (6-3 e x)^{5/2}\right ) \, dx\\ &=-\frac {24 \sqrt {3} (2-e x)^{5/2}}{5 e}+\frac {6 \sqrt {3} (2-e x)^{7/2}}{7 e}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 43, normalized size = 0.96 \[ -\frac {6 (e x-2)^2 (5 e x+18) \sqrt {12-3 e^2 x^2}}{35 e \sqrt {e x+2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.99, size = 54, normalized size = 1.20 \[ -\frac {6 \, {\left (5 \, e^{3} x^{3} - 2 \, e^{2} x^{2} - 52 \, e x + 72\right )} \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2}}{35 \, {\left (e^{2} x + 2 \, e\right )}} \]
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 \[ \int \frac {{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {3}{2}}}{\sqrt {e x + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 36, normalized size = 0.80 \[ \frac {2 \left (e x -2\right ) \left (5 e x +18\right ) \left (-3 e^{2} x^{2}+12\right )^{\frac {3}{2}}}{35 \left (e x +2\right )^{\frac {3}{2}} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 2.97, size = 47, normalized size = 1.04 \[ -\frac {{\left (30 i \, \sqrt {3} e^{3} x^{3} - 12 i \, \sqrt {3} e^{2} x^{2} - 312 i \, \sqrt {3} e x + 432 i \, \sqrt {3}\right )} \sqrt {e x - 2}}{35 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 43, normalized size = 0.96 \[ \frac {\sqrt {12-3\,e^2\,x^2}\,\left (\frac {312\,x}{35}+\frac {12\,e\,x^2}{35}-\frac {432}{35\,e}-\frac {6\,e^2\,x^3}{7}\right )}{\sqrt {e\,x+2}} \]
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 \[ 3 \sqrt {3} \left (\int \frac {4 \sqrt {- e^{2} x^{2} + 4}}{\sqrt {e x + 2}}\, dx + \int \left (- \frac {e^{2} x^{2} \sqrt {- e^{2} x^{2} + 4}}{\sqrt {e x + 2}}\right )\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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