Optimal. Leaf size=65 \[ -\frac {2 (2-e x)^{9/2}}{\sqrt {3} e}+\frac {48 \sqrt {3} (2-e x)^{7/2}}{7 e}-\frac {96 \sqrt {3} (2-e x)^{5/2}}{5 e} \]
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Rubi [A] time = 0.02, antiderivative size = 65, 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} \[ -\frac {2 (2-e x)^{9/2}}{\sqrt {3} e}+\frac {48 \sqrt {3} (2-e x)^{7/2}}{7 e}-\frac {96 \sqrt {3} (2-e x)^{5/2}}{5 e} \]
Antiderivative was successfully verified.
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Rule 43
Rule 627
Rubi steps
\begin {align*} \int \sqrt {2+e x} \left (12-3 e^2 x^2\right )^{3/2} \, dx &=\int (6-3 e x)^{3/2} (2+e x)^2 \, dx\\ &=\int \left (16 (6-3 e x)^{3/2}-\frac {8}{3} (6-3 e x)^{5/2}+\frac {1}{9} (6-3 e x)^{7/2}\right ) \, dx\\ &=-\frac {96 \sqrt {3} (2-e x)^{5/2}}{5 e}+\frac {48 \sqrt {3} (2-e x)^{7/2}}{7 e}-\frac {2 (2-e x)^{9/2}}{\sqrt {3} e}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 52, normalized size = 0.80 \[ -\frac {2 (e x-2)^2 \sqrt {4-e^2 x^2} \left (35 e^2 x^2+220 e x+428\right )}{35 e \sqrt {3 e x+6}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 62, normalized size = 0.95 \[ -\frac {2 \, {\left (35 \, e^{4} x^{4} + 80 \, e^{3} x^{3} - 312 \, e^{2} x^{2} - 832 \, e x + 1712\right )} \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2}}{105 \, {\left (e^{2} x + 2 \, e\right )}} \]
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 = 0.05, size = 44, normalized size = 0.68 \[ \frac {2 \left (e x -2\right ) \left (35 e^{2} x^{2}+220 e x +428\right ) \left (-3 e^{2} x^{2}+12\right )^{\frac {3}{2}}}{315 \left (e x +2\right )^{\frac {3}{2}} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 3.18, size = 71, normalized size = 1.09 \[ -\frac {{\left (70 i \, \sqrt {3} e^{4} x^{4} + 160 i \, \sqrt {3} e^{3} x^{3} - 624 i \, \sqrt {3} e^{2} x^{2} - 1664 i \, \sqrt {3} e x + 3424 i \, \sqrt {3}\right )} {\left (e x + 2\right )} \sqrt {e x - 2}}{105 \, {\left (e^{2} x + 2 \, e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 71, normalized size = 1.09 \[ \frac {2\,\sqrt {12-3\,e^2\,x^2}\,\sqrt {e\,x+2}\,\left (-35\,e^3\,x^3-10\,e^2\,x^2+332\,e\,x+168\right )}{105\,e}-\frac {4096\,\sqrt {12-3\,e^2\,x^2}}{105\,e\,\sqrt {e\,x+2}} \]
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 \[ 3 \sqrt {3} \left (\int 4 \sqrt {e x + 2} \sqrt {- e^{2} x^{2} + 4}\, dx + \int \left (- e^{2} x^{2} \sqrt {e x + 2} \sqrt {- e^{2} x^{2} + 4}\right )\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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