Integrand size = 24, antiderivative size = 46 \[ \int \frac {\sqrt {12-3 e^2 x^2}}{(2+e x)^{3/2}} \, dx=\frac {2 \sqrt {3} \sqrt {2-e x}}{e}-\frac {4 \sqrt {3} \text {arctanh}\left (\frac {1}{2} \sqrt {2-e x}\right )}{e} \]
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Time = 0.01 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {641, 52, 65, 212} \[ \int \frac {\sqrt {12-3 e^2 x^2}}{(2+e x)^{3/2}} \, dx=\frac {2 \sqrt {3} \sqrt {2-e x}}{e}-\frac {4 \sqrt {3} \text {arctanh}\left (\frac {1}{2} \sqrt {2-e x}\right )}{e} \]
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Rule 52
Rule 65
Rule 212
Rule 641
Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt {6-3 e x}}{2+e x} \, dx \\ & = \frac {2 \sqrt {3} \sqrt {2-e x}}{e}+12 \int \frac {1}{\sqrt {6-3 e x} (2+e x)} \, dx \\ & = \frac {2 \sqrt {3} \sqrt {2-e x}}{e}-\frac {8 \text {Subst}\left (\int \frac {1}{4-\frac {x^2}{3}} \, dx,x,\sqrt {6-3 e x}\right )}{e} \\ & = \frac {2 \sqrt {3} \sqrt {2-e x}}{e}-\frac {4 \sqrt {3} \tanh ^{-1}\left (\frac {1}{2} \sqrt {2-e x}\right )}{e} \\ \end{align*}
Time = 0.36 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.37 \[ \int \frac {\sqrt {12-3 e^2 x^2}}{(2+e x)^{3/2}} \, dx=\frac {2 \sqrt {3} \left (\frac {\sqrt {4-e^2 x^2}}{\sqrt {2+e x}}-2 \text {arctanh}\left (\frac {2 \sqrt {2+e x}}{\sqrt {4-e^2 x^2}}\right )\right )}{e} \]
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Time = 2.22 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.43
method | result | size |
default | \(-\frac {2 \sqrt {-x^{2} e^{2}+4}\, \left (2 \sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {-3 e x +6}\, \sqrt {3}}{6}\right )-\sqrt {-3 e x +6}\right ) \sqrt {3}}{\sqrt {e x +2}\, \sqrt {-3 e x +6}\, e}\) | \(66\) |
risch | \(-\frac {6 \left (e x -2\right ) \sqrt {\frac {-3 x^{2} e^{2}+12}{e x +2}}\, \sqrt {e x +2}}{e \sqrt {-3 e x +6}\, \sqrt {-3 x^{2} e^{2}+12}}-\frac {4 \sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {-3 e x +6}\, \sqrt {3}}{6}\right ) \sqrt {\frac {-3 x^{2} e^{2}+12}{e x +2}}\, \sqrt {e x +2}}{e \sqrt {-3 x^{2} e^{2}+12}}\) | \(120\) |
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Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (36) = 72\).
Time = 0.28 (sec) , antiderivative size = 99, normalized size of antiderivative = 2.15 \[ \int \frac {\sqrt {12-3 e^2 x^2}}{(2+e x)^{3/2}} \, dx=\frac {2 \, {\left (\sqrt {3} {\left (e x + 2\right )} \log \left (-\frac {3 \, e^{2} x^{2} - 12 \, e x + 4 \, \sqrt {3} \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2} - 36}{e^{2} x^{2} + 4 \, e x + 4}\right ) + \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2}\right )}}{e^{2} x + 2 \, e} \]
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\[ \int \frac {\sqrt {12-3 e^2 x^2}}{(2+e x)^{3/2}} \, dx=\sqrt {3} \int \frac {\sqrt {- e^{2} x^{2} + 4}}{e x \sqrt {e x + 2} + 2 \sqrt {e x + 2}}\, dx \]
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\[ \int \frac {\sqrt {12-3 e^2 x^2}}{(2+e x)^{3/2}} \, dx=\int { \frac {\sqrt {-3 \, e^{2} x^{2} + 12}}{{\left (e x + 2\right )}^{\frac {3}{2}}} \,d x } \]
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Exception generated. \[ \int \frac {\sqrt {12-3 e^2 x^2}}{(2+e x)^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\sqrt {12-3 e^2 x^2}}{(2+e x)^{3/2}} \, dx=\int \frac {\sqrt {12-3\,e^2\,x^2}}{{\left (e\,x+2\right )}^{3/2}} \,d x \]
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