Optimal. Leaf size=65 \[ -\frac {2 (2-e x)^{5/2}}{5 \sqrt {3} e}+\frac {16 (2-e x)^{3/2}}{3 \sqrt {3} e}-\frac {32 \sqrt {2-e x}}{\sqrt {3} e} \]
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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} \begin {gather*} -\frac {2 (2-e x)^{5/2}}{5 \sqrt {3} e}+\frac {16 (2-e x)^{3/2}}{3 \sqrt {3} e}-\frac {32 \sqrt {2-e x}}{\sqrt {3} e} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 627
Rubi steps
\begin {align*} \int \frac {(2+e x)^{5/2}}{\sqrt {12-3 e^2 x^2}} \, dx &=\int \frac {(2+e x)^2}{\sqrt {6-3 e x}} \, dx\\ &=\int \left (\frac {16}{\sqrt {6-3 e x}}-\frac {8}{3} \sqrt {6-3 e x}+\frac {1}{9} (6-3 e x)^{3/2}\right ) \, dx\\ &=-\frac {32 \sqrt {2-e x}}{\sqrt {3} e}+\frac {16 (2-e x)^{3/2}}{3 \sqrt {3} e}-\frac {2 (2-e x)^{5/2}}{5 \sqrt {3} e}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 49, normalized size = 0.75 \begin {gather*} \frac {2 (e x-2) \sqrt {e x+2} \left (3 e^2 x^2+28 e x+172\right )}{15 e \sqrt {12-3 e^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.32, size = 71, normalized size = 1.09 \begin {gather*} -\frac {2 \sqrt {4 (e x+2)-(e x+2)^2} \left (3 \sqrt {3} (e x+2)^2+16 \sqrt {3} (e x+2)+128 \sqrt {3}\right )}{45 e \sqrt {e x+2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 46, normalized size = 0.71 \begin {gather*} -\frac {2 \, {\left (3 \, e^{2} x^{2} + 28 \, e x + 172\right )} \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2}}{45 \, {\left (e^{2} x + 2 \, 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 [A] time = 0.05, size = 44, normalized size = 0.68 \begin {gather*} \frac {2 \left (e x -2\right ) \left (3 e^{2} x^{2}+28 e x +172\right ) \sqrt {e x +2}}{15 \sqrt {-3 e^{2} x^{2}+12}\, e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 3.04, size = 47, normalized size = 0.72 \begin {gather*} -\frac {6 i \, \sqrt {3} e^{3} x^{3} + 44 i \, \sqrt {3} e^{2} x^{2} + 232 i \, \sqrt {3} e x - 688 i \, \sqrt {3}}{45 \, \sqrt {e x - 2} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.55, size = 61, normalized size = 0.94 \begin {gather*} -\frac {\sqrt {12-3\,e^2\,x^2}\,\left (\frac {344\,\sqrt {e\,x+2}}{45\,e^2}+\frac {2\,x^2\,\sqrt {e\,x+2}}{15}+\frac {56\,x\,\sqrt {e\,x+2}}{45\,e}\right )}{x+\frac {2}{e}} \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} \left (\int \frac {4 \sqrt {e x + 2}}{\sqrt {- e^{2} x^{2} + 4}}\, dx + \int \frac {4 e x \sqrt {e x + 2}}{\sqrt {- e^{2} x^{2} + 4}}\, dx + \int \frac {e^{2} x^{2} \sqrt {e x + 2}}{\sqrt {- e^{2} x^{2} + 4}}\, dx\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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