Optimal. Leaf size=65 \[ -\frac {32 \sqrt {2-e x}}{\sqrt {3} e}+\frac {16 (2-e x)^{3/2}}{3 \sqrt {3} e}-\frac {2 (2-e x)^{5/2}}{5 \sqrt {3} e} \]
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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 = {641, 45}
\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 45
Rule 641
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 = 0.22, size = 45, normalized size = 0.69 \begin {gather*} -\frac {2 \sqrt {4-e^2 x^2} \left (172+28 e x+3 e^2 x^2\right )}{15 e \sqrt {6+3 e x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.47, size = 39, normalized size = 0.60
method | result | size |
default | \(-\frac {2 \sqrt {-3 e^{2} x^{2}+12}\, \left (3 e^{2} x^{2}+28 e x +172\right )}{45 \sqrt {e x +2}\, e}\) | \(39\) |
gosper | \(\frac {2 \left (e x -2\right ) \left (3 e^{2} x^{2}+28 e x +172\right ) \sqrt {e x +2}}{15 e \sqrt {-3 e^{2} x^{2}+12}}\) | \(44\) |
risch | \(\frac {2 \sqrt {\frac {-3 e^{2} x^{2}+12}{e x +2}}\, \sqrt {e x +2}\, \left (3 e^{2} x^{2}+28 e x +172\right ) \left (e x -2\right )}{15 \sqrt {-3 e^{2} x^{2}+12}\, e \sqrt {-3 e x +6}}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.51, size = 46, normalized size = 0.71 \begin {gather*} \frac {2 \, {\left (-3 i \, \sqrt {3} x^{3} e^{3} - 22 i \, \sqrt {3} x^{2} e^{2} - 116 i \, \sqrt {3} x e + 344 i \, \sqrt {3}\right )} e^{\left (-1\right )}}{45 \, \sqrt {x e - 2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.15, size = 46, normalized size = 0.71 \begin {gather*} -\frac {2 \, {\left (3 \, x^{2} e^{2} + 28 \, x e + 172\right )} \sqrt {-3 \, x^{2} e^{2} + 12} \sqrt {x e + 2}}{45 \, {\left (x e^{2} + 2 \, e\right )}} \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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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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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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