Optimal. Leaf size=65 \[ -\frac {32 (2-e x)^{3/2}}{\sqrt {3} e}+\frac {16 \sqrt {3} (2-e x)^{5/2}}{5 e}-\frac {2 \sqrt {3} (2-e x)^{7/2}}{7 e} \]
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Rubi [A]
time = 0.01, 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 \sqrt {3} (2-e x)^{7/2}}{7 e}+\frac {16 \sqrt {3} (2-e x)^{5/2}}{5 e}-\frac {32 (2-e x)^{3/2}}{\sqrt {3} e} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 641
Rubi steps
\begin {align*} \int (2+e x)^{3/2} \sqrt {12-3 e^2 x^2} \, dx &=\int \sqrt {6-3 e x} (2+e x)^2 \, dx\\ &=\int \left (16 \sqrt {6-3 e x}-\frac {8}{3} (6-3 e x)^{3/2}+\frac {1}{9} (6-3 e x)^{5/2}\right ) \, dx\\ &=-\frac {32 (2-e x)^{3/2}}{\sqrt {3} e}+\frac {16 \sqrt {3} (2-e x)^{5/2}}{5 e}-\frac {2 \sqrt {3} (2-e x)^{7/2}}{7 e}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 50, normalized size = 0.77 \begin {gather*} \frac {2 (-2+e x) \sqrt {4-e^2 x^2} \left (284+108 e x+15 e^2 x^2\right )}{35 e \sqrt {6+3 e x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.47, size = 44, normalized size = 0.68
method | result | size |
gosper | \(\frac {2 \left (e x -2\right ) \left (15 e^{2} x^{2}+108 e x +284\right ) \sqrt {-3 e^{2} x^{2}+12}}{105 \sqrt {e x +2}\, e}\) | \(44\) |
default | \(\frac {2 \left (e x -2\right ) \left (15 e^{2} x^{2}+108 e x +284\right ) \sqrt {-3 e^{2} x^{2}+12}}{105 \sqrt {e x +2}\, e}\) | \(44\) |
risch | \(-\frac {2 \sqrt {\frac {-3 e^{2} x^{2}+12}{e x +2}}\, \sqrt {e x +2}\, \left (15 e^{3} x^{3}+78 e^{2} x^{2}+68 e x -568\right ) \left (e x -2\right )}{35 \sqrt {-3 e^{2} x^{2}+12}\, e \sqrt {-3 e x +6}}\) | \(80\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.55, size = 61, normalized size = 0.94 \begin {gather*} -\frac {2 \, {\left (-15 i \, \sqrt {3} x^{3} e^{3} - 78 i \, \sqrt {3} x^{2} e^{2} - 68 i \, \sqrt {3} x e + 568 i \, \sqrt {3}\right )} {\left (x e + 2\right )} \sqrt {x e - 2}}{105 \, {\left (x e^{2} + 2 \, e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.78, size = 53, normalized size = 0.82 \begin {gather*} \frac {2 \, {\left (15 \, x^{3} e^{3} + 78 \, x^{2} e^{2} + 68 \, x e - 568\right )} \sqrt {-3 \, x^{2} e^{2} + 12} \sqrt {x e + 2}}{105 \, {\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*} \sqrt {3} \left (\int 2 \sqrt {e x + 2} \sqrt {- e^{2} x^{2} + 4}\, dx + \int e x \sqrt {e x + 2} \sqrt {- e^{2} x^{2} + 4}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 100 vs.
\(2 (49) = 98\).
time = 1.70, size = 100, normalized size = 1.54 \begin {gather*} \frac {2}{105} \, \sqrt {3} {\left (84 \, {\left (x e - 2\right )}^{2} \sqrt {-x e + 2} + {\left ({\left (15 \, {\left (x e - 2\right )}^{3} \sqrt {-x e + 2} + 84 \, {\left (x e - 2\right )}^{2} \sqrt {-x e + 2} - 140 \, {\left (-x e + 2\right )}^{\frac {3}{2}}\right )} e^{\left (-2\right )} + 352 \, e^{\left (-2\right )}\right )} e^{2} - 420 \, {\left (-x e + 2\right )}^{\frac {3}{2}} + 672\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.52, size = 73, normalized size = 1.12 \begin {gather*} \frac {\sqrt {12-3\,e^2\,x^2}\,\left (\frac {52\,x^2\,\sqrt {e\,x+2}}{35}-\frac {1136\,\sqrt {e\,x+2}}{105\,e^2}+\frac {136\,x\,\sqrt {e\,x+2}}{105\,e}+\frac {2\,e\,x^3\,\sqrt {e\,x+2}}{7}\right )}{x+\frac {2}{e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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