Integrand size = 24, antiderivative size = 108 \[ \int \frac {1}{(2+e x)^{3/2} \left (12-3 e^2 x^2\right )^{3/2}} \, dx=\frac {5}{256 \sqrt {3} e \sqrt {2-e x}}-\frac {1}{24 \sqrt {3} e \sqrt {2-e x} (2+e x)^2}-\frac {5}{192 \sqrt {3} e \sqrt {2-e x} (2+e x)}-\frac {5 \text {arctanh}\left (\frac {1}{2} \sqrt {2-e x}\right )}{512 \sqrt {3} e} \]
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Time = 0.03 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {641, 44, 53, 65, 212} \[ \int \frac {1}{(2+e x)^{3/2} \left (12-3 e^2 x^2\right )^{3/2}} \, dx=-\frac {5 \text {arctanh}\left (\frac {1}{2} \sqrt {2-e x}\right )}{512 \sqrt {3} e}+\frac {5}{256 \sqrt {3} e \sqrt {2-e x}}-\frac {5}{192 \sqrt {3} e \sqrt {2-e x} (e x+2)}-\frac {1}{24 \sqrt {3} e \sqrt {2-e x} (e x+2)^2} \]
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Rule 44
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)^3} \, dx \\ & = -\frac {1}{24 \sqrt {3} e \sqrt {2-e x} (2+e x)^2}+\frac {5}{16} \int \frac {1}{(6-3 e x)^{3/2} (2+e x)^2} \, dx \\ & = -\frac {1}{24 \sqrt {3} e \sqrt {2-e x} (2+e x)^2}-\frac {5}{192 \sqrt {3} e \sqrt {2-e x} (2+e x)}+\frac {15}{128} \int \frac {1}{(6-3 e x)^{3/2} (2+e x)} \, dx \\ & = \frac {5}{256 \sqrt {3} e \sqrt {2-e x}}-\frac {1}{24 \sqrt {3} e \sqrt {2-e x} (2+e x)^2}-\frac {5}{192 \sqrt {3} e \sqrt {2-e x} (2+e x)}+\frac {5}{512} \int \frac {1}{\sqrt {6-3 e x} (2+e x)} \, dx \\ & = \frac {5}{256 \sqrt {3} e \sqrt {2-e x}}-\frac {1}{24 \sqrt {3} e \sqrt {2-e x} (2+e x)^2}-\frac {5}{192 \sqrt {3} e \sqrt {2-e x} (2+e x)}-\frac {5 \text {Subst}\left (\int \frac {1}{4-\frac {x^2}{3}} \, dx,x,\sqrt {6-3 e x}\right )}{768 e} \\ & = \frac {5}{256 \sqrt {3} e \sqrt {2-e x}}-\frac {1}{24 \sqrt {3} e \sqrt {2-e x} (2+e x)^2}-\frac {5}{192 \sqrt {3} e \sqrt {2-e x} (2+e x)}-\frac {5 \tanh ^{-1}\left (\frac {1}{2} \sqrt {2-e x}\right )}{512 \sqrt {3} e} \\ \end{align*}
Time = 0.61 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.81 \[ \int \frac {1}{(2+e x)^{3/2} \left (12-3 e^2 x^2\right )^{3/2}} \, dx=\frac {-\frac {2 \sqrt {4-e^2 x^2} \left (-12+40 e x+15 e^2 x^2\right )}{(-2+e x) (2+e x)^{5/2}}-15 \text {arctanh}\left (\frac {2 \sqrt {2+e x}}{\sqrt {4-e^2 x^2}}\right )}{1536 \sqrt {3} e} \]
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Time = 2.32 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.25
method | result | size |
default | \(\frac {\sqrt {-3 x^{2} e^{2}+12}\, \left (5 \sqrt {3}\, \sqrt {-3 e x +6}\, \operatorname {arctanh}\left (\frac {\sqrt {-3 e x +6}\, \sqrt {3}}{6}\right ) e^{2} x^{2}+20 \sqrt {3}\, \sqrt {-3 e x +6}\, \operatorname {arctanh}\left (\frac {\sqrt {-3 e x +6}\, \sqrt {3}}{6}\right ) e x -30 x^{2} e^{2}+20 \sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {-3 e x +6}\, \sqrt {3}}{6}\right ) \sqrt {-3 e x +6}-80 e x +24\right )}{4608 \left (e x +2\right )^{\frac {5}{2}} \left (e x -2\right ) e}\) | \(135\) |
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Time = 0.38 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.36 \[ \int \frac {1}{(2+e x)^{3/2} \left (12-3 e^2 x^2\right )^{3/2}} \, dx=\frac {15 \, \sqrt {3} {\left (e^{4} x^{4} + 4 \, e^{3} x^{3} - 16 \, e x - 16\right )} \log \left (-\frac {3 \, e^{2} x^{2} - 12 \, e x + 4 \, \sqrt {3} \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2} - 36}{e^{2} x^{2} + 4 \, e x + 4}\right ) - 4 \, {\left (15 \, e^{2} x^{2} + 40 \, e x - 12\right )} \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2}}{9216 \, {\left (e^{5} x^{4} + 4 \, e^{4} x^{3} - 16 \, e^{2} x - 16 \, e\right )}} \]
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\[ \int \frac {1}{(2+e x)^{3/2} \left (12-3 e^2 x^2\right )^{3/2}} \, dx=\frac {\sqrt {3} \int \frac {1}{- e^{3} x^{3} \sqrt {e x + 2} \sqrt {- e^{2} x^{2} + 4} - 2 e^{2} x^{2} \sqrt {e x + 2} \sqrt {- e^{2} x^{2} + 4} + 4 e x \sqrt {e x + 2} \sqrt {- e^{2} x^{2} + 4} + 8 \sqrt {e x + 2} \sqrt {- e^{2} x^{2} + 4}}\, dx}{9} \]
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\[ \int \frac {1}{(2+e x)^{3/2} \left (12-3 e^2 x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {3}{2}} {\left (e x + 2\right )}^{\frac {3}{2}}} \,d x } \]
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Exception generated. \[ \int \frac {1}{(2+e x)^{3/2} \left (12-3 e^2 x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {1}{(2+e x)^{3/2} \left (12-3 e^2 x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (12-3\,e^2\,x^2\right )}^{3/2}\,{\left (e\,x+2\right )}^{3/2}} \,d x \]
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