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 = {641, 32}
\begin {gather*} -\frac {6 \sqrt {3} (2-e x)^{5/2}}{5 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 641
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 = 0.18, size = 35, normalized size = 1.59 \begin {gather*} -\frac {6 \sqrt {3} \left (4-e^2 x^2\right )^{5/2}}{5 e (2+e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.47, size = 32, normalized size = 1.45
method | result | size |
gosper | \(\frac {2 \left (e x -2\right ) \left (-3 e^{2} x^{2}+12\right )^{\frac {3}{2}}}{5 e \left (e x +2\right )^{\frac {3}{2}}}\) | \(30\) |
default | \(-\frac {6 \sqrt {-3 e^{2} x^{2}+12}\, \left (e x -2\right )^{2}}{5 \sqrt {e x +2}\, e}\) | \(32\) |
risch | \(\frac {18 \sqrt {\frac {-3 e^{2} x^{2}+12}{e x +2}}\, \sqrt {e x +2}\, \left (e^{2} x^{2}-4 e x +4\right ) \left (e x -2\right )}{5 \sqrt {-3 e^{2} x^{2}+12}\, e \sqrt {-3 e x +6}}\) | \(71\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.50, size = 36, normalized size = 1.64 \begin {gather*} \frac {6}{5} \, {\left (-i \, \sqrt {3} x^{2} e^{2} + 4 i \, \sqrt {3} x e - 4 i \, \sqrt {3}\right )} \sqrt {x e - 2} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 45 vs.
\(2 (16) = 32\).
time = 2.30, size = 45, normalized size = 2.05 \begin {gather*} -\frac {6 \, {\left (x^{2} e^{2} - 4 \, x e + 4\right )} \sqrt {-3 \, x^{2} e^{2} + 12} \sqrt {x e + 2}}{5 \, {\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*} 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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.84, size = 27, normalized size = 1.23 \begin {gather*} -\frac {6}{5} \, \sqrt {3} {\left ({\left (x e - 2\right )}^{2} \sqrt {-x e + 2} - 32\right )} e^{\left (-1\right )} \end {gather*}
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 \begin {gather*} -\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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