Optimal. Leaf size=87 \[ -\frac {384 \sqrt {3} (2-e x)^{5/2}}{5 e}+\frac {288 \sqrt {3} (2-e x)^{7/2}}{7 e}-\frac {8 \sqrt {3} (2-e x)^{9/2}}{e}+\frac {6 \sqrt {3} (2-e x)^{11/2}}{11 e} \]
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Rubi [A]
time = 0.02, 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 = {641, 45}
\begin {gather*} \frac {6 \sqrt {3} (2-e x)^{11/2}}{11 e}-\frac {8 \sqrt {3} (2-e x)^{9/2}}{e}+\frac {288 \sqrt {3} (2-e x)^{7/2}}{7 e}-\frac {384 \sqrt {3} (2-e x)^{5/2}}{5 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} \left (12-3 e^2 x^2\right )^{3/2} \, dx &=\int (6-3 e x)^{3/2} (2+e x)^3 \, dx\\ &=\int \left (64 (6-3 e x)^{3/2}-16 (6-3 e x)^{5/2}+\frac {4}{3} (6-3 e x)^{7/2}-\frac {1}{27} (6-3 e x)^{9/2}\right ) \, dx\\ &=-\frac {384 \sqrt {3} (2-e x)^{5/2}}{5 e}+\frac {288 \sqrt {3} (2-e x)^{7/2}}{7 e}-\frac {8 \sqrt {3} (2-e x)^{9/2}}{e}+\frac {6 \sqrt {3} (2-e x)^{11/2}}{11 e}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 59, normalized size = 0.68 \begin {gather*} -\frac {2 (-2+e x)^2 \sqrt {12-3 e^2 x^2} \left (4264+3020 e x+910 e^2 x^2+105 e^3 x^3\right )}{385 e \sqrt {2+e x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.48, size = 54, normalized size = 0.62
method | result | size |
gosper | \(\frac {2 \left (e x -2\right ) \left (105 e^{3} x^{3}+910 e^{2} x^{2}+3020 e x +4264\right ) \left (-3 e^{2} x^{2}+12\right )^{\frac {3}{2}}}{1155 e \left (e x +2\right )^{\frac {3}{2}}}\) | \(52\) |
default | \(-\frac {2 \sqrt {-3 e^{2} x^{2}+12}\, \left (e x -2\right )^{2} \left (105 e^{3} x^{3}+910 e^{2} x^{2}+3020 e x +4264\right )}{385 \sqrt {e x +2}\, e}\) | \(54\) |
risch | \(\frac {6 \sqrt {\frac {-3 e^{2} x^{2}+12}{e x +2}}\, \sqrt {e x +2}\, \left (105 e^{5} x^{5}+490 e^{4} x^{4}-200 e^{3} x^{3}-4176 e^{2} x^{2}-4976 e x +17056\right ) \left (e x -2\right )}{385 \sqrt {-3 e^{2} x^{2}+12}\, e \sqrt {-3 e x +6}}\) | \(96\) |
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 = 81, normalized size = 0.93 \begin {gather*} \frac {2 \, {\left (-105 i \, \sqrt {3} x^{5} e^{5} - 490 i \, \sqrt {3} x^{4} e^{4} + 200 i \, \sqrt {3} x^{3} e^{3} + 4176 i \, \sqrt {3} x^{2} e^{2} + 4976 i \, \sqrt {3} x e - 17056 i \, \sqrt {3}\right )} {\left (x e + 2\right )} \sqrt {x e - 2}}{385 \, {\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 = 2.71, size = 67, normalized size = 0.77 \begin {gather*} -\frac {2 \, {\left (105 \, x^{5} e^{5} + 490 \, x^{4} e^{4} - 200 \, x^{3} e^{3} - 4176 \, x^{2} e^{2} - 4976 \, x e + 17056\right )} \sqrt {-3 \, x^{2} e^{2} + 12} \sqrt {x e + 2}}{385 \, {\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 8 \sqrt {e x + 2} \sqrt {- e^{2} x^{2} + 4}\, dx + \int 4 e x \sqrt {e x + 2} \sqrt {- e^{2} x^{2} + 4}\, dx + \int \left (- 2 e^{2} x^{2} \sqrt {e x + 2} \sqrt {- e^{2} x^{2} + 4}\right )\, dx + \int \left (- e^{3} x^{3} \sqrt {e x + 2} \sqrt {- e^{2} x^{2} + 4}\right )\, 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. 220 vs.
\(2 (65) = 130\).
time = 1.11, size = 220, normalized size = 2.53 \begin {gather*} \frac {2}{1155} \, \sqrt {3} {\left (11088 \, {\left (x e - 2\right )}^{2} \sqrt {-x e + 2} - {\left ({\left (315 \, {\left (x e - 2\right )}^{5} \sqrt {-x e + 2} + 3080 \, {\left (x e - 2\right )}^{4} \sqrt {-x e + 2} + 11880 \, {\left (x e - 2\right )}^{3} \sqrt {-x e + 2} + 22176 \, {\left (x e - 2\right )}^{2} \sqrt {-x e + 2} - 18480 \, {\left (-x e + 2\right )}^{\frac {3}{2}}\right )} e^{\left (-4\right )} + 27008 \, e^{\left (-4\right )}\right )} e^{4} - 44 \, {\left ({\left (35 \, {\left (x e - 2\right )}^{4} \sqrt {-x e + 2} + 270 \, {\left (x e - 2\right )}^{3} \sqrt {-x e + 2} + 756 \, {\left (x e - 2\right )}^{2} \sqrt {-x e + 2} - 840 \, {\left (-x e + 2\right )}^{\frac {3}{2}}\right )} e^{\left (-3\right )} - 832 \, e^{\left (-3\right )}\right )} e^{3} - 55440 \, {\left (-x e + 2\right )}^{\frac {3}{2}} + 88704\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.59, size = 53, normalized size = 0.61 \begin {gather*} -\frac {2\,\sqrt {12-3\,e^2\,x^2}\,{\left (e\,x-2\right )}^2\,\left (105\,e^3\,x^3+910\,e^2\,x^2+3020\,e\,x+4264\right )}{385\,e\,\sqrt {e\,x+2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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