Optimal. Leaf size=22 \[ -\frac {6 \sqrt {3} (2-e x)^{5/2}}{5 e} \]
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Rubi [A] time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {627, 32} \[ -\frac {6 \sqrt {3} (2-e x)^{5/2}}{5 e} \]
Antiderivative was successfully verified.
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Rule 32
Rule 627
Rubi steps
\begin {align*} \int \frac {\left (12-3 e^2 x^2\right )^{3/2}}{(2+e x)^{3/2}} \, dx &=\int (6-3 e x)^{3/2} \, dx\\ &=-\frac {6 \sqrt {3} (2-e x)^{5/2}}{5 e}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 37, normalized size = 1.68 \[ -\frac {6 (e x-2)^2 \sqrt {12-3 e^2 x^2}}{5 e \sqrt {e x+2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.98, size = 45, normalized size = 2.05 \[ -\frac {6 \, {\left (e^{2} x^{2} - 4 \, e x + 4\right )} \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2}}{5 \, {\left (e^{2} x + 2 \, e\right )}} \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 30, normalized size = 1.36 \[ \frac {2 \left (e x -2\right ) \left (-3 e^{2} x^{2}+12\right )^{\frac {3}{2}}}{5 \left (e x +2\right )^{\frac {3}{2}} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 3.09, size = 36, normalized size = 1.64 \[ -\frac {{\left (6 i \, \sqrt {3} e^{2} x^{2} - 24 i \, \sqrt {3} e x + 24 i \, \sqrt {3}\right )} \sqrt {e x - 2}}{5 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.50, size = 36, normalized size = 1.64 \[ -\frac {\sqrt {12-3\,e^2\,x^2}\,\left (\frac {6\,e\,x^2}{5}-\frac {24\,x}{5}+\frac {24}{5\,e}\right )}{\sqrt {e\,x+2}} \]
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 \[ 3 \sqrt {3} \left (\int \frac {4 \sqrt {- e^{2} x^{2} + 4}}{e x \sqrt {e x + 2} + 2 \sqrt {e x + 2}}\, dx + \int \left (- \frac {e^{2} x^{2} \sqrt {- e^{2} x^{2} + 4}}{e x \sqrt {e x + 2} + 2 \sqrt {e x + 2}}\right )\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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