Integrand size = 24, antiderivative size = 65 \[ \int \frac {(2+e x)^{5/2}}{\sqrt {12-3 e^2 x^2}} \, dx=-\frac {32 \sqrt {2-e x}}{\sqrt {3} e}+\frac {16 (2-e x)^{3/2}}{3 \sqrt {3} e}-\frac {2 (2-e x)^{5/2}}{5 \sqrt {3} e} \]
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Time = 0.02 (sec) , antiderivative size = 65, 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} \[ \int \frac {(2+e x)^{5/2}}{\sqrt {12-3 e^2 x^2}} \, dx=-\frac {2 (2-e x)^{5/2}}{5 \sqrt {3} e}+\frac {16 (2-e x)^{3/2}}{3 \sqrt {3} e}-\frac {32 \sqrt {2-e x}}{\sqrt {3} e} \]
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Rule 45
Rule 641
Rubi steps \begin{align*} \text {integral}& = \int \frac {(2+e x)^2}{\sqrt {6-3 e x}} \, dx \\ & = \int \left (\frac {16}{\sqrt {6-3 e x}}-\frac {8}{3} \sqrt {6-3 e x}+\frac {1}{9} (6-3 e x)^{3/2}\right ) \, dx \\ & = -\frac {32 \sqrt {2-e x}}{\sqrt {3} e}+\frac {16 (2-e x)^{3/2}}{3 \sqrt {3} e}-\frac {2 (2-e x)^{5/2}}{5 \sqrt {3} e} \\ \end{align*}
Time = 0.39 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.69 \[ \int \frac {(2+e x)^{5/2}}{\sqrt {12-3 e^2 x^2}} \, dx=-\frac {2 \sqrt {4-e^2 x^2} \left (172+28 e x+3 e^2 x^2\right )}{15 e \sqrt {6+3 e x}} \]
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Time = 2.24 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.60
method | result | size |
default | \(-\frac {2 \sqrt {-3 x^{2} e^{2}+12}\, \left (3 x^{2} e^{2}+28 e x +172\right )}{45 \sqrt {e x +2}\, e}\) | \(39\) |
gosper | \(\frac {2 \left (e x -2\right ) \left (3 x^{2} e^{2}+28 e x +172\right ) \sqrt {e x +2}}{15 e \sqrt {-3 x^{2} e^{2}+12}}\) | \(44\) |
risch | \(\frac {2 \sqrt {\frac {-3 x^{2} e^{2}+12}{e x +2}}\, \sqrt {e x +2}\, \left (3 x^{2} e^{2}+28 e x +172\right ) \left (e x -2\right )}{15 \sqrt {-3 x^{2} e^{2}+12}\, e \sqrt {-3 e x +6}}\) | \(72\) |
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none
Time = 0.38 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.71 \[ \int \frac {(2+e x)^{5/2}}{\sqrt {12-3 e^2 x^2}} \, dx=-\frac {2 \, {\left (3 \, e^{2} x^{2} + 28 \, e x + 172\right )} \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2}}{45 \, {\left (e^{2} x + 2 \, e\right )}} \]
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\[ \int \frac {(2+e x)^{5/2}}{\sqrt {12-3 e^2 x^2}} \, dx=\frac {\sqrt {3} \left (\int \frac {4 \sqrt {e x + 2}}{\sqrt {- e^{2} x^{2} + 4}}\, dx + \int \frac {4 e x \sqrt {e x + 2}}{\sqrt {- e^{2} x^{2} + 4}}\, dx + \int \frac {e^{2} x^{2} \sqrt {e x + 2}}{\sqrt {- e^{2} x^{2} + 4}}\, dx\right )}{3} \]
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Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.72 \[ \int \frac {(2+e x)^{5/2}}{\sqrt {12-3 e^2 x^2}} \, dx=\frac {2 \, {\left (-3 i \, \sqrt {3} e^{3} x^{3} - 22 i \, \sqrt {3} e^{2} x^{2} - 116 i \, \sqrt {3} e x + 344 i \, \sqrt {3}\right )}}{45 \, \sqrt {e x - 2} e} \]
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Exception generated. \[ \int \frac {(2+e x)^{5/2}}{\sqrt {12-3 e^2 x^2}} \, dx=\text {Exception raised: TypeError} \]
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Time = 10.09 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.94 \[ \int \frac {(2+e x)^{5/2}}{\sqrt {12-3 e^2 x^2}} \, dx=-\frac {\sqrt {12-3\,e^2\,x^2}\,\left (\frac {344\,\sqrt {e\,x+2}}{45\,e^2}+\frac {2\,x^2\,\sqrt {e\,x+2}}{15}+\frac {56\,x\,\sqrt {e\,x+2}}{45\,e}\right )}{x+\frac {2}{e}} \]
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