Integrand size = 24, antiderivative size = 43 \[ \int \frac {(2+e x)^{3/2}}{\sqrt {12-3 e^2 x^2}} \, dx=-\frac {8 \sqrt {2-e x}}{\sqrt {3} e}+\frac {2 (2-e x)^{3/2}}{3 \sqrt {3} e} \]
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Time = 0.01 (sec) , antiderivative size = 43, 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} \[ \int \frac {(2+e x)^{3/2}}{\sqrt {12-3 e^2 x^2}} \, dx=\frac {2 (2-e x)^{3/2}}{3 \sqrt {3} e}-\frac {8 \sqrt {2-e x}}{\sqrt {3} e} \]
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Rule 45
Rule 641
Rubi steps \begin{align*} \text {integral}& = \int \frac {2+e x}{\sqrt {6-3 e x}} \, dx \\ & = \int \left (\frac {4}{\sqrt {6-3 e x}}-\frac {1}{3} \sqrt {6-3 e x}\right ) \, dx \\ & = -\frac {8 \sqrt {2-e x}}{\sqrt {3} e}+\frac {2 (2-e x)^{3/2}}{3 \sqrt {3} e} \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.84 \[ \int \frac {(2+e x)^{3/2}}{\sqrt {12-3 e^2 x^2}} \, dx=-\frac {2 (10+e x) \sqrt {4-e^2 x^2}}{3 e \sqrt {6+3 e x}} \]
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Time = 2.31 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.70
method | result | size |
default | \(-\frac {2 \sqrt {-3 x^{2} e^{2}+12}\, \left (e x +10\right )}{9 \sqrt {e x +2}\, e}\) | \(30\) |
gosper | \(\frac {2 \left (e x -2\right ) \left (e x +10\right ) \sqrt {e x +2}}{3 e \sqrt {-3 x^{2} e^{2}+12}}\) | \(35\) |
risch | \(\frac {2 \sqrt {\frac {-3 x^{2} e^{2}+12}{e x +2}}\, \sqrt {e x +2}\, \left (e x +10\right ) \left (e x -2\right )}{3 \sqrt {-3 x^{2} e^{2}+12}\, e \sqrt {-3 e x +6}}\) | \(63\) |
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none
Time = 0.39 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.86 \[ \int \frac {(2+e x)^{3/2}}{\sqrt {12-3 e^2 x^2}} \, dx=-\frac {2 \, \sqrt {-3 \, e^{2} x^{2} + 12} {\left (e x + 10\right )} \sqrt {e x + 2}}{9 \, {\left (e^{2} x + 2 \, e\right )}} \]
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\[ \int \frac {(2+e x)^{3/2}}{\sqrt {12-3 e^2 x^2}} \, dx=\frac {\sqrt {3} \left (\int \frac {2 \sqrt {e x + 2}}{\sqrt {- e^{2} x^{2} + 4}}\, dx + \int \frac {e x \sqrt {e x + 2}}{\sqrt {- e^{2} x^{2} + 4}}\, dx\right )}{3} \]
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.65 \[ \int \frac {(2+e x)^{3/2}}{\sqrt {12-3 e^2 x^2}} \, dx=-\frac {2 i \, \sqrt {3} {\left (e^{2} x^{2} + 8 \, e x - 20\right )}}{9 \, \sqrt {e x - 2} e} \]
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Exception generated. \[ \int \frac {(2+e x)^{3/2}}{\sqrt {12-3 e^2 x^2}} \, dx=\text {Exception raised: TypeError} \]
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Time = 10.21 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.14 \[ \int \frac {(2+e x)^{3/2}}{\sqrt {12-3 e^2 x^2}} \, dx=-\frac {\left (\frac {20\,\sqrt {e\,x+2}}{9\,e^2}+\frac {2\,x\,\sqrt {e\,x+2}}{9\,e}\right )\,\sqrt {12-3\,e^2\,x^2}}{x+\frac {2}{e}} \]
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