Integrand size = 24, antiderivative size = 66 \[ \int \frac {\left (12-3 e^2 x^2\right )^{3/2}}{(2+e x)^{5/2}} \, dx=\frac {24 \sqrt {3} \sqrt {2-e x}}{e}+\frac {2 \sqrt {3} (2-e x)^{3/2}}{e}-\frac {48 \sqrt {3} \text {arctanh}\left (\frac {1}{2} \sqrt {2-e x}\right )}{e} \]
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Time = 0.02 (sec) , antiderivative size = 66, 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, 52, 65, 212} \[ \int \frac {\left (12-3 e^2 x^2\right )^{3/2}}{(2+e x)^{5/2}} \, dx=-\frac {48 \sqrt {3} \text {arctanh}\left (\frac {1}{2} \sqrt {2-e x}\right )}{e}+\frac {2 \sqrt {3} (2-e x)^{3/2}}{e}+\frac {24 \sqrt {3} \sqrt {2-e x}}{e} \]
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Rule 52
Rule 65
Rule 212
Rule 641
Rubi steps \begin{align*} \text {integral}& = \int \frac {(6-3 e x)^{3/2}}{2+e x} \, dx \\ & = \frac {2 \sqrt {3} (2-e x)^{3/2}}{e}+12 \int \frac {\sqrt {6-3 e x}}{2+e x} \, dx \\ & = \frac {24 \sqrt {3} \sqrt {2-e x}}{e}+\frac {2 \sqrt {3} (2-e x)^{3/2}}{e}+144 \int \frac {1}{\sqrt {6-3 e x} (2+e x)} \, dx \\ & = \frac {24 \sqrt {3} \sqrt {2-e x}}{e}+\frac {2 \sqrt {3} (2-e x)^{3/2}}{e}-\frac {96 \text {Subst}\left (\int \frac {1}{4-\frac {x^2}{3}} \, dx,x,\sqrt {6-3 e x}\right )}{e} \\ & = \frac {24 \sqrt {3} \sqrt {2-e x}}{e}+\frac {2 \sqrt {3} (2-e x)^{3/2}}{e}-\frac {48 \sqrt {3} \tanh ^{-1}\left (\frac {1}{2} \sqrt {2-e x}\right )}{e} \\ \end{align*}
Time = 0.46 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.17 \[ \int \frac {\left (12-3 e^2 x^2\right )^{3/2}}{(2+e x)^{5/2}} \, dx=-\frac {2 \sqrt {3} \left ((-14+e x) \sqrt {4-e^2 x^2}+24 \sqrt {2+e x} \text {arctanh}\left (\frac {2 \sqrt {2+e x}}{\sqrt {4-e^2 x^2}}\right )\right )}{e \sqrt {2+e x}} \]
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Time = 2.27 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.17
method | result | size |
default | \(-\frac {2 \sqrt {-x^{2} e^{2}+4}\, \left (e x \sqrt {-3 e x +6}+24 \sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {-3 e x +6}\, \sqrt {3}}{6}\right )-14 \sqrt {-3 e x +6}\right ) \sqrt {3}}{\sqrt {e x +2}\, \sqrt {-3 e x +6}\, e}\) | \(77\) |
risch | \(\frac {6 \left (e x -14\right ) \left (e x -2\right ) \sqrt {\frac {-3 x^{2} e^{2}+12}{e x +2}}\, \sqrt {e x +2}}{e \sqrt {-3 e x +6}\, \sqrt {-3 x^{2} e^{2}+12}}-\frac {48 \sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {-3 e x +6}\, \sqrt {3}}{6}\right ) \sqrt {\frac {-3 x^{2} e^{2}+12}{e x +2}}\, \sqrt {e x +2}}{e \sqrt {-3 x^{2} e^{2}+12}}\) | \(125\) |
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Leaf count of result is larger than twice the leaf count of optimal. 106 vs. \(2 (52) = 104\).
Time = 0.29 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.61 \[ \int \frac {\left (12-3 e^2 x^2\right )^{3/2}}{(2+e x)^{5/2}} \, dx=\frac {2 \, {\left (12 \, \sqrt {3} {\left (e x + 2\right )} \log \left (-\frac {3 \, e^{2} x^{2} - 12 \, e x + 4 \, \sqrt {3} \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2} - 36}{e^{2} x^{2} + 4 \, e x + 4}\right ) - \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2} {\left (e x - 14\right )}\right )}}{e^{2} x + 2 \, e} \]
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\[ \int \frac {\left (12-3 e^2 x^2\right )^{3/2}}{(2+e x)^{5/2}} \, dx=3 \sqrt {3} \left (\int \frac {4 \sqrt {- e^{2} x^{2} + 4}}{e^{2} x^{2} \sqrt {e x + 2} + 4 e x \sqrt {e x + 2} + 4 \sqrt {e x + 2}}\, dx + \int \left (- \frac {e^{2} x^{2} \sqrt {- e^{2} x^{2} + 4}}{e^{2} x^{2} \sqrt {e x + 2} + 4 e x \sqrt {e x + 2} + 4 \sqrt {e x + 2}}\right )\, dx\right ) \]
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\[ \int \frac {\left (12-3 e^2 x^2\right )^{3/2}}{(2+e x)^{5/2}} \, dx=\int { \frac {{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {3}{2}}}{{\left (e x + 2\right )}^{\frac {5}{2}}} \,d x } \]
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none
Time = 0.28 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.83 \[ \int \frac {\left (12-3 e^2 x^2\right )^{3/2}}{(2+e x)^{5/2}} \, dx=\frac {2 \, \sqrt {3} {\left ({\left (-e x + 2\right )}^{\frac {3}{2}} + 12 \, \sqrt {-e x + 2} - 12 \, \log \left (\sqrt {-e x + 2} + 2\right ) + 12 \, \log \left (-\sqrt {-e x + 2} + 2\right )\right )}}{e} \]
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Timed out. \[ \int \frac {\left (12-3 e^2 x^2\right )^{3/2}}{(2+e x)^{5/2}} \, dx=\int \frac {{\left (12-3\,e^2\,x^2\right )}^{3/2}}{{\left (e\,x+2\right )}^{5/2}} \,d x \]
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