Integrand size = 24, antiderivative size = 86 \[ \int \frac {1}{(2+e x)^{5/2} \sqrt {12-3 e^2 x^2}} \, dx=-\frac {\sqrt {2-e x}}{8 \sqrt {3} e (2+e x)^2}-\frac {\sqrt {3} \sqrt {2-e x}}{64 e (2+e x)}-\frac {\sqrt {3} \text {arctanh}\left (\frac {1}{2} \sqrt {2-e x}\right )}{128 e} \]
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Time = 0.02 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {641, 44, 65, 212} \[ \int \frac {1}{(2+e x)^{5/2} \sqrt {12-3 e^2 x^2}} \, dx=-\frac {\sqrt {3} \text {arctanh}\left (\frac {1}{2} \sqrt {2-e x}\right )}{128 e}-\frac {\sqrt {3} \sqrt {2-e x}}{64 e (e x+2)}-\frac {\sqrt {2-e x}}{8 \sqrt {3} e (e x+2)^2} \]
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Rule 44
Rule 65
Rule 212
Rule 641
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\sqrt {6-3 e x} (2+e x)^3} \, dx \\ & = -\frac {\sqrt {2-e x}}{8 \sqrt {3} e (2+e x)^2}+\frac {3}{16} \int \frac {1}{\sqrt {6-3 e x} (2+e x)^2} \, dx \\ & = -\frac {\sqrt {2-e x}}{8 \sqrt {3} e (2+e x)^2}-\frac {\sqrt {3} \sqrt {2-e x}}{64 e (2+e x)}+\frac {3}{128} \int \frac {1}{\sqrt {6-3 e x} (2+e x)} \, dx \\ & = -\frac {\sqrt {2-e x}}{8 \sqrt {3} e (2+e x)^2}-\frac {\sqrt {3} \sqrt {2-e x}}{64 e (2+e x)}-\frac {\text {Subst}\left (\int \frac {1}{4-\frac {x^2}{3}} \, dx,x,\sqrt {6-3 e x}\right )}{64 e} \\ & = -\frac {\sqrt {2-e x}}{8 \sqrt {3} e (2+e x)^2}-\frac {\sqrt {3} \sqrt {2-e x}}{64 e (2+e x)}-\frac {\sqrt {3} \tanh ^{-1}\left (\frac {1}{2} \sqrt {2-e x}\right )}{128 e} \\ \end{align*}
Time = 0.49 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.94 \[ \int \frac {1}{(2+e x)^{5/2} \sqrt {12-3 e^2 x^2}} \, dx=\frac {-2 (14+3 e x) \sqrt {4-e^2 x^2}-3 (2+e x)^{5/2} \text {arctanh}\left (\frac {2 \sqrt {2+e x}}{\sqrt {4-e^2 x^2}}\right )}{128 \sqrt {3} e (2+e x)^{5/2}} \]
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Time = 2.30 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.44
method | result | size |
default | \(-\frac {\sqrt {-x^{2} e^{2}+4}\, \left (3 \sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {-3 e x +6}\, \sqrt {3}}{6}\right ) e^{2} x^{2}+12 \sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {-3 e x +6}\, \sqrt {3}}{6}\right ) e x +6 e x \sqrt {-3 e x +6}+12 \sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {-3 e x +6}\, \sqrt {3}}{6}\right )+28 \sqrt {-3 e x +6}\right ) \sqrt {3}}{384 \left (e x +2\right )^{\frac {5}{2}} \sqrt {-3 e x +6}\, e}\) | \(124\) |
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Leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (66) = 132\).
Time = 0.29 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.62 \[ \int \frac {1}{(2+e x)^{5/2} \sqrt {12-3 e^2 x^2}} \, dx=\frac {3 \, \sqrt {3} {\left (e^{3} x^{3} + 6 \, e^{2} x^{2} + 12 \, e x + 8\right )} \log \left (-\frac {3 \, e^{2} x^{2} - 12 \, e x + 4 \, \sqrt {3} \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2} - 36}{e^{2} x^{2} + 4 \, e x + 4}\right ) - 4 \, \sqrt {-3 \, e^{2} x^{2} + 12} {\left (3 \, e x + 14\right )} \sqrt {e x + 2}}{768 \, {\left (e^{4} x^{3} + 6 \, e^{3} x^{2} + 12 \, e^{2} x + 8 \, e\right )}} \]
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\[ \int \frac {1}{(2+e x)^{5/2} \sqrt {12-3 e^2 x^2}} \, dx=\frac {\sqrt {3} \int \frac {1}{e^{2} x^{2} \sqrt {e x + 2} \sqrt {- e^{2} x^{2} + 4} + 4 e x \sqrt {e x + 2} \sqrt {- e^{2} x^{2} + 4} + 4 \sqrt {e x + 2} \sqrt {- e^{2} x^{2} + 4}}\, dx}{3} \]
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\[ \int \frac {1}{(2+e x)^{5/2} \sqrt {12-3 e^2 x^2}} \, dx=\int { \frac {1}{\sqrt {-3 \, e^{2} x^{2} + 12} {\left (e x + 2\right )}^{\frac {5}{2}}} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.78 \[ \int \frac {1}{(2+e x)^{5/2} \sqrt {12-3 e^2 x^2}} \, dx=\frac {\sqrt {3} {\left (\frac {4 \, {\left (3 \, {\left (-e x + 2\right )}^{\frac {3}{2}} - 20 \, \sqrt {-e x + 2}\right )}}{{\left (e x + 2\right )}^{2}} - 3 \, \log \left (\sqrt {-e x + 2} + 2\right ) + 3 \, \log \left (-\sqrt {-e x + 2} + 2\right )\right )}}{768 \, e} \]
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Timed out. \[ \int \frac {1}{(2+e x)^{5/2} \sqrt {12-3 e^2 x^2}} \, dx=\int \frac {1}{\sqrt {12-3\,e^2\,x^2}\,{\left (e\,x+2\right )}^{5/2}} \,d x \]
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