Integrand size = 24, antiderivative size = 50 \[ \int \frac {\sqrt {2+e x}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=\frac {1}{6 \sqrt {3} e \sqrt {2-e x}}-\frac {\text {arctanh}\left (\frac {1}{2} \sqrt {2-e x}\right )}{12 \sqrt {3} e} \]
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Time = 0.01 (sec) , antiderivative size = 50, 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, 53, 65, 212} \[ \int \frac {\sqrt {2+e x}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=\frac {1}{6 \sqrt {3} e \sqrt {2-e x}}-\frac {\text {arctanh}\left (\frac {1}{2} \sqrt {2-e x}\right )}{12 \sqrt {3} e} \]
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Rule 53
Rule 65
Rule 212
Rule 641
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(6-3 e x)^{3/2} (2+e x)} \, dx \\ & = \frac {1}{6 \sqrt {3} e \sqrt {2-e x}}+\frac {1}{12} \int \frac {1}{\sqrt {6-3 e x} (2+e x)} \, dx \\ & = \frac {1}{6 \sqrt {3} e \sqrt {2-e x}}-\frac {\text {Subst}\left (\int \frac {1}{4-\frac {x^2}{3}} \, dx,x,\sqrt {6-3 e x}\right )}{18 e} \\ & = \frac {1}{6 \sqrt {3} e \sqrt {2-e x}}-\frac {\tanh ^{-1}\left (\frac {1}{2} \sqrt {2-e x}\right )}{12 \sqrt {3} e} \\ \end{align*}
Time = 0.44 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.66 \[ \int \frac {\sqrt {2+e x}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=\frac {-2 \sqrt {4-e^2 x^2}-(-2+e x) \sqrt {2+e x} \text {arctanh}\left (\frac {2 \sqrt {2+e x}}{\sqrt {4-e^2 x^2}}\right )}{12 e (-2+e x) \sqrt {6+3 e x}} \]
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Time = 2.64 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.20
method | result | size |
default | \(\frac {\sqrt {-3 x^{2} e^{2}+12}\, \left (\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {-3 e x +6}\, \sqrt {3}}{6}\right ) \sqrt {-3 e x +6}-6\right )}{108 \sqrt {e x +2}\, \left (e x -2\right ) e}\) | \(60\) |
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Leaf count of result is larger than twice the leaf count of optimal. 106 vs. \(2 (36) = 72\).
Time = 0.39 (sec) , antiderivative size = 106, normalized size of antiderivative = 2.12 \[ \int \frac {\sqrt {2+e x}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=\frac {\sqrt {3} {\left (e^{2} x^{2} - 4\right )} \log \left (-\frac {3 \, e^{2} x^{2} - 12 \, e x + 4 \, \sqrt {3} \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2} - 36}{e^{2} x^{2} + 4 \, e x + 4}\right ) - 4 \, \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2}}{72 \, {\left (e^{3} x^{2} - 4 \, e\right )}} \]
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\[ \int \frac {\sqrt {2+e x}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=\frac {\sqrt {3} \int \frac {\sqrt {e x + 2}}{- e^{2} x^{2} \sqrt {- e^{2} x^{2} + 4} + 4 \sqrt {- e^{2} x^{2} + 4}}\, dx}{9} \]
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\[ \int \frac {\sqrt {2+e x}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {e x + 2}}{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {3}{2}}} \,d x } \]
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none
Time = 0.27 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.14 \[ \int \frac {\sqrt {2+e x}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=-\frac {\sqrt {3} \log \left (\sqrt {-e x + 2} + 2\right )}{72 \, e} + \frac {\sqrt {3} \log \left (-\sqrt {-e x + 2} + 2\right )}{72 \, e} + \frac {\sqrt {3}}{18 \, \sqrt {-e x + 2} e} \]
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Timed out. \[ \int \frac {\sqrt {2+e x}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {e\,x+2}}{{\left (12-3\,e^2\,x^2\right )}^{3/2}} \,d x \]
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