Optimal. Leaf size=87 \[ \frac {2 (2-e x)^{9/2}}{3 \sqrt {3} e}-\frac {24 \sqrt {3} (2-e x)^{7/2}}{7 e}+\frac {96 \sqrt {3} (2-e x)^{5/2}}{5 e}-\frac {128 (2-e x)^{3/2}}{\sqrt {3} e} \]
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Rubi [A] time = 0.03, antiderivative size = 87, 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} \[ \frac {2 (2-e x)^{9/2}}{3 \sqrt {3} e}-\frac {24 \sqrt {3} (2-e x)^{7/2}}{7 e}+\frac {96 \sqrt {3} (2-e x)^{5/2}}{5 e}-\frac {128 (2-e x)^{3/2}}{\sqrt {3} e} \]
Antiderivative was successfully verified.
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Rule 43
Rule 627
Rubi steps
\begin {align*} \int (2+e x)^{5/2} \sqrt {12-3 e^2 x^2} \, dx &=\int \sqrt {6-3 e x} (2+e x)^3 \, dx\\ &=\int \left (64 \sqrt {6-3 e x}-16 (6-3 e x)^{3/2}+\frac {4}{3} (6-3 e x)^{5/2}-\frac {1}{27} (6-3 e x)^{7/2}\right ) \, dx\\ &=-\frac {128 (2-e x)^{3/2}}{\sqrt {3} e}+\frac {96 \sqrt {3} (2-e x)^{5/2}}{5 e}-\frac {24 \sqrt {3} (2-e x)^{7/2}}{7 e}+\frac {2 (2-e x)^{9/2}}{3 \sqrt {3} e}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 58, normalized size = 0.67 \[ \frac {2 (e x-2) \sqrt {4-e^2 x^2} \left (35 e^3 x^3+330 e^2 x^2+1284 e x+2552\right )}{105 e \sqrt {3 e x+6}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.03, size = 62, normalized size = 0.71 \[ \frac {2 \, {\left (35 \, e^{4} x^{4} + 260 \, e^{3} x^{3} + 624 \, e^{2} x^{2} - 16 \, e x - 5104\right )} \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2}}{315 \, {\left (e^{2} x + 2 \, e\right )}} \]
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 = 52, normalized size = 0.60 \[ \frac {2 \left (e x -2\right ) \left (35 e^{3} x^{3}+330 e^{2} x^{2}+1284 e x +2552\right ) \sqrt {-3 e^{2} x^{2}+12}}{315 \sqrt {e x +2}\, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 3.13, size = 71, normalized size = 0.82 \[ \frac {{\left (70 i \, \sqrt {3} e^{4} x^{4} + 520 i \, \sqrt {3} e^{3} x^{3} + 1248 i \, \sqrt {3} e^{2} x^{2} - 32 i \, \sqrt {3} e x - 10208 i \, \sqrt {3}\right )} {\left (e x + 2\right )} \sqrt {e x - 2}}{315 \, {\left (e^{2} x + 2 \, e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.31, size = 54, normalized size = 0.62 \[ \frac {2\,\sqrt {12-3\,e^2\,x^2}\,\left (35\,e^4\,x^4+260\,e^3\,x^3+624\,e^2\,x^2-16\,e\,x-5104\right )}{315\,e\,\sqrt {e\,x+2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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