Integrand size = 24, antiderivative size = 67 \[ \int \frac {(2+e x)^{7/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=\frac {32}{3 \sqrt {3} e \sqrt {2-e x}}+\frac {16 \sqrt {2-e x}}{3 \sqrt {3} e}-\frac {2 (2-e x)^{3/2}}{9 \sqrt {3} e} \]
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Time = 0.01 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {641, 45} \[ \int \frac {(2+e x)^{7/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=-\frac {2 (2-e x)^{3/2}}{9 \sqrt {3} e}+\frac {16 \sqrt {2-e x}}{3 \sqrt {3} e}+\frac {32}{3 \sqrt {3} e \sqrt {2-e x}} \]
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Rule 45
Rule 641
Rubi steps \begin{align*} \text {integral}& = \int \frac {(2+e x)^2}{(6-3 e x)^{3/2}} \, dx \\ & = \int \left (\frac {16}{(6-3 e x)^{3/2}}-\frac {8}{3 \sqrt {6-3 e x}}+\frac {1}{9} \sqrt {6-3 e x}\right ) \, dx \\ & = \frac {32}{3 \sqrt {3} e \sqrt {2-e x}}+\frac {16 \sqrt {2-e x}}{3 \sqrt {3} e}-\frac {2 (2-e x)^{3/2}}{9 \sqrt {3} e} \\ \end{align*}
Time = 0.49 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.76 \[ \int \frac {(2+e x)^{7/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=\frac {2 \sqrt {4-e^2 x^2} \left (-92+20 e x+e^2 x^2\right )}{9 e (-2+e x) \sqrt {6+3 e x}} \]
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Time = 2.42 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.64
method | result | size |
gosper | \(\frac {2 \left (e x -2\right ) \left (x^{2} e^{2}+20 e x -92\right ) \left (e x +2\right )^{\frac {3}{2}}}{3 e \left (-3 x^{2} e^{2}+12\right )^{\frac {3}{2}}}\) | \(43\) |
default | \(\frac {2 \sqrt {-3 x^{2} e^{2}+12}\, \left (x^{2} e^{2}+20 e x -92\right )}{27 \sqrt {e x +2}\, \left (e x -2\right ) e}\) | \(45\) |
risch | \(-\frac {2 \left (e x +22\right ) \left (e x -2\right ) \sqrt {\frac {-3 x^{2} e^{2}+12}{e x +2}}\, \sqrt {e x +2}}{9 e \sqrt {-3 e x +6}\, \sqrt {-3 x^{2} e^{2}+12}}+\frac {32 \sqrt {\frac {-3 x^{2} e^{2}+12}{e x +2}}\, \sqrt {e x +2}}{3 e \sqrt {-3 e x +6}\, \sqrt {-3 x^{2} e^{2}+12}}\) | \(116\) |
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none
Time = 0.29 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.70 \[ \int \frac {(2+e x)^{7/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=\frac {2 \, {\left (e^{2} x^{2} + 20 \, e x - 92\right )} \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2}}{27 \, {\left (e^{3} x^{2} - 4 \, e\right )}} \]
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Timed out. \[ \int \frac {(2+e x)^{7/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=\text {Timed out} \]
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.54 \[ \int \frac {(2+e x)^{7/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=-\frac {2 \, {\left (-i \, \sqrt {3} e^{2} x^{2} - 20 i \, \sqrt {3} e x + 92 i \, \sqrt {3}\right )}}{27 \, \sqrt {e x - 2} e} \]
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Exception generated. \[ \int \frac {(2+e x)^{7/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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Time = 10.25 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.72 \[ \int \frac {(2+e x)^{7/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=\frac {2\,\sqrt {12-3\,e^2\,x^2}\,\sqrt {e\,x+2}\,\left (e^2\,x^2+20\,e\,x-92\right )}{27\,e\,\left (e^2\,x^2-4\right )} \]
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