Integrand size = 24, antiderivative size = 86 \[ \int \frac {\left (12-3 e^2 x^2\right )^{3/2}}{(2+e x)^{9/2}} \, dx=-\frac {3 \sqrt {3} (2-e x)^{3/2}}{2 e (2+e x)^2}+\frac {9 \sqrt {3} \sqrt {2-e x}}{4 e (2+e x)}-\frac {9 \sqrt {3} \text {arctanh}\left (\frac {1}{2} \sqrt {2-e x}\right )}{8 e} \]
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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, 43, 65, 212} \[ \int \frac {\left (12-3 e^2 x^2\right )^{3/2}}{(2+e x)^{9/2}} \, dx=-\frac {9 \sqrt {3} \text {arctanh}\left (\frac {1}{2} \sqrt {2-e x}\right )}{8 e}-\frac {3 \sqrt {3} (2-e x)^{3/2}}{2 e (e x+2)^2}+\frac {9 \sqrt {3} \sqrt {2-e x}}{4 e (e x+2)} \]
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Rule 43
Rule 65
Rule 212
Rule 641
Rubi steps \begin{align*} \text {integral}& = \int \frac {(6-3 e x)^{3/2}}{(2+e x)^3} \, dx \\ & = -\frac {3 \sqrt {3} (2-e x)^{3/2}}{2 e (2+e x)^2}-\frac {9}{4} \int \frac {\sqrt {6-3 e x}}{(2+e x)^2} \, dx \\ & = -\frac {3 \sqrt {3} (2-e x)^{3/2}}{2 e (2+e x)^2}+\frac {9 \sqrt {3} \sqrt {2-e x}}{4 e (2+e x)}+\frac {27}{8} \int \frac {1}{\sqrt {6-3 e x} (2+e x)} \, dx \\ & = -\frac {3 \sqrt {3} (2-e x)^{3/2}}{2 e (2+e x)^2}+\frac {9 \sqrt {3} \sqrt {2-e x}}{4 e (2+e x)}-\frac {9 \text {Subst}\left (\int \frac {1}{4-\frac {x^2}{3}} \, dx,x,\sqrt {6-3 e x}\right )}{4 e} \\ & = -\frac {3 \sqrt {3} (2-e x)^{3/2}}{2 e (2+e x)^2}+\frac {9 \sqrt {3} \sqrt {2-e x}}{4 e (2+e x)}-\frac {9 \sqrt {3} \tanh ^{-1}\left (\frac {1}{2} \sqrt {2-e x}\right )}{8 e} \\ \end{align*}
Time = 0.52 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.84 \[ \int \frac {\left (12-3 e^2 x^2\right )^{3/2}}{(2+e x)^{9/2}} \, dx=\frac {3 \sqrt {3} \left (\frac {2 (2+5 e x) \sqrt {4-e^2 x^2}}{(2+e x)^{5/2}}-3 \text {arctanh}\left (\frac {2 \sqrt {2+e x}}{\sqrt {4-e^2 x^2}}\right )\right )}{8 e} \]
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Time = 2.24 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.44
method | result | size |
default | \(-\frac {3 \sqrt {-x^{2} e^{2}+4}\, \left (3 \sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {-3 e x +6}\, \sqrt {3}}{6}\right ) e^{2} x^{2}+12 \sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {-3 e x +6}\, \sqrt {3}}{6}\right ) e x -10 e x \sqrt {-3 e x +6}+12 \sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {-3 e x +6}\, \sqrt {3}}{6}\right )-4 \sqrt {-3 e x +6}\right ) \sqrt {3}}{8 \left (e x +2\right )^{\frac {5}{2}} \sqrt {-3 e x +6}\, e}\) | \(124\) |
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Leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (66) = 132\).
Time = 0.35 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.62 \[ \int \frac {\left (12-3 e^2 x^2\right )^{3/2}}{(2+e x)^{9/2}} \, dx=\frac {3 \, {\left (3 \, \sqrt {3} {\left (e^{3} x^{3} + 6 \, e^{2} x^{2} + 12 \, e x + 8\right )} \log \left (-\frac {3 \, e^{2} x^{2} - 12 \, e x + 4 \, \sqrt {3} \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2} - 36}{e^{2} x^{2} + 4 \, e x + 4}\right ) + 4 \, \sqrt {-3 \, e^{2} x^{2} + 12} {\left (5 \, e x + 2\right )} \sqrt {e x + 2}\right )}}{16 \, {\left (e^{4} x^{3} + 6 \, e^{3} x^{2} + 12 \, e^{2} x + 8 \, e\right )}} \]
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Timed out. \[ \int \frac {\left (12-3 e^2 x^2\right )^{3/2}}{(2+e x)^{9/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (12-3 e^2 x^2\right )^{3/2}}{(2+e x)^{9/2}} \, dx=\int { \frac {{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {3}{2}}}{{\left (e x + 2\right )}^{\frac {9}{2}}} \,d x } \]
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none
Time = 0.28 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.78 \[ \int \frac {\left (12-3 e^2 x^2\right )^{3/2}}{(2+e x)^{9/2}} \, dx=-\frac {3 \, \sqrt {3} {\left (\frac {4 \, {\left (5 \, {\left (-e x + 2\right )}^{\frac {3}{2}} - 12 \, \sqrt {-e x + 2}\right )}}{{\left (e x + 2\right )}^{2}} + 3 \, \log \left (\sqrt {-e x + 2} + 2\right ) - 3 \, \log \left (-\sqrt {-e x + 2} + 2\right )\right )}}{16 \, e} \]
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Timed out. \[ \int \frac {\left (12-3 e^2 x^2\right )^{3/2}}{(2+e x)^{9/2}} \, dx=\int \frac {{\left (12-3\,e^2\,x^2\right )}^{3/2}}{{\left (e\,x+2\right )}^{9/2}} \,d x \]
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