Integrand size = 24, antiderivative size = 45 \[ \int \frac {\left (12-3 e^2 x^2\right )^{3/2}}{\sqrt {2+e x}} \, dx=-\frac {24 \sqrt {3} (2-e x)^{5/2}}{5 e}+\frac {6 \sqrt {3} (2-e x)^{7/2}}{7 e} \]
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Time = 0.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {641, 45} \[ \int \frac {\left (12-3 e^2 x^2\right )^{3/2}}{\sqrt {2+e x}} \, dx=\frac {6 \sqrt {3} (2-e x)^{7/2}}{7 e}-\frac {24 \sqrt {3} (2-e x)^{5/2}}{5 e} \]
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Rule 45
Rule 641
Rubi steps \begin{align*} \text {integral}& = \int (6-3 e x)^{3/2} (2+e x) \, dx \\ & = \int \left (4 (6-3 e x)^{3/2}-\frac {1}{3} (6-3 e x)^{5/2}\right ) \, dx \\ & = -\frac {24 \sqrt {3} (2-e x)^{5/2}}{5 e}+\frac {6 \sqrt {3} (2-e x)^{7/2}}{7 e} \\ \end{align*}
Time = 0.32 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.96 \[ \int \frac {\left (12-3 e^2 x^2\right )^{3/2}}{\sqrt {2+e x}} \, dx=-\frac {6 (-2+e x)^2 (18+5 e x) \sqrt {12-3 e^2 x^2}}{35 e \sqrt {2+e x}} \]
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Time = 2.31 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.80
method | result | size |
gosper | \(\frac {2 \left (e x -2\right ) \left (5 e x +18\right ) \left (-3 x^{2} e^{2}+12\right )^{\frac {3}{2}}}{35 e \left (e x +2\right )^{\frac {3}{2}}}\) | \(36\) |
default | \(-\frac {6 \sqrt {-3 x^{2} e^{2}+12}\, \left (e x -2\right )^{2} \left (5 e x +18\right )}{35 \sqrt {e x +2}\, e}\) | \(38\) |
risch | \(\frac {18 \sqrt {\frac {-3 x^{2} e^{2}+12}{e x +2}}\, \sqrt {e x +2}\, \left (5 e^{3} x^{3}-2 x^{2} e^{2}-52 e x +72\right ) \left (e x -2\right )}{35 \sqrt {-3 x^{2} e^{2}+12}\, e \sqrt {-3 e x +6}}\) | \(80\) |
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none
Time = 0.28 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.20 \[ \int \frac {\left (12-3 e^2 x^2\right )^{3/2}}{\sqrt {2+e x}} \, dx=-\frac {6 \, {\left (5 \, e^{3} x^{3} - 2 \, e^{2} x^{2} - 52 \, e x + 72\right )} \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2}}{35 \, {\left (e^{2} x + 2 \, e\right )}} \]
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\[ \int \frac {\left (12-3 e^2 x^2\right )^{3/2}}{\sqrt {2+e x}} \, dx=3 \sqrt {3} \left (\int \frac {4 \sqrt {- e^{2} x^{2} + 4}}{\sqrt {e x + 2}}\, dx + \int \left (- \frac {e^{2} x^{2} \sqrt {- e^{2} x^{2} + 4}}{\sqrt {e x + 2}}\right )\, dx\right ) \]
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.04 \[ \int \frac {\left (12-3 e^2 x^2\right )^{3/2}}{\sqrt {2+e x}} \, dx=\frac {6 \, {\left (-5 i \, \sqrt {3} e^{3} x^{3} + 2 i \, \sqrt {3} e^{2} x^{2} + 52 i \, \sqrt {3} e x - 72 i \, \sqrt {3}\right )} \sqrt {e x - 2}}{35 \, e} \]
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Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (33) = 66\).
Time = 0.28 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.76 \[ \int \frac {\left (12-3 e^2 x^2\right )^{3/2}}{\sqrt {2+e x}} \, dx=-\frac {2 \, \sqrt {3} {\left (e^{2} {\left (\frac {15 \, {\left (e x - 2\right )}^{3} \sqrt {-e x + 2} + 84 \, {\left (e x - 2\right )}^{2} \sqrt {-e x + 2} - 140 \, {\left (-e x + 2\right )}^{\frac {3}{2}}}{e^{2}} + \frac {352}{e^{2}}\right )} + 140 \, {\left (-e x + 2\right )}^{\frac {3}{2}} - 1120\right )}}{35 \, e} \]
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Time = 0.16 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.96 \[ \int \frac {\left (12-3 e^2 x^2\right )^{3/2}}{\sqrt {2+e x}} \, dx=\frac {\sqrt {12-3\,e^2\,x^2}\,\left (\frac {312\,x}{35}+\frac {12\,e\,x^2}{35}-\frac {432}{35\,e}-\frac {6\,e^2\,x^3}{7}\right )}{\sqrt {e\,x+2}} \]
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