Integrand size = 24, antiderivative size = 35 \[ \int \frac {1}{(2+e x)^{3/2} \sqrt [4]{12-3 e^2 x^2}} \, dx=-\frac {\left (4-e^2 x^2\right )^{3/4}}{3 \sqrt [4]{3} e (2+e x)^{3/2}} \]
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Time = 0.01 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {665} \[ \int \frac {1}{(2+e x)^{3/2} \sqrt [4]{12-3 e^2 x^2}} \, dx=-\frac {\left (4-e^2 x^2\right )^{3/4}}{3 \sqrt [4]{3} e (e x+2)^{3/2}} \]
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Rule 665
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (4-e^2 x^2\right )^{3/4}}{3 \sqrt [4]{3} e (2+e x)^{3/2}} \\ \end{align*}
Time = 0.41 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(2+e x)^{3/2} \sqrt [4]{12-3 e^2 x^2}} \, dx=-\frac {\left (4-e^2 x^2\right )^{3/4}}{3 \sqrt [4]{3} e (2+e x)^{3/2}} \]
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Time = 2.33 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.86
method | result | size |
gosper | \(\frac {e x -2}{3 \sqrt {e x +2}\, e \left (-3 x^{2} e^{2}+12\right )^{\frac {1}{4}}}\) | \(30\) |
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none
Time = 0.64 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.14 \[ \int \frac {1}{(2+e x)^{3/2} \sqrt [4]{12-3 e^2 x^2}} \, dx=-\frac {{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {3}{4}} \sqrt {e x + 2}}{9 \, {\left (e^{3} x^{2} + 4 \, e^{2} x + 4 \, e\right )}} \]
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\[ \int \frac {1}{(2+e x)^{3/2} \sqrt [4]{12-3 e^2 x^2}} \, dx=\frac {3^{\frac {3}{4}} \int \frac {1}{e x \sqrt {e x + 2} \sqrt [4]{- e^{2} x^{2} + 4} + 2 \sqrt {e x + 2} \sqrt [4]{- e^{2} x^{2} + 4}}\, dx}{3} \]
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\[ \int \frac {1}{(2+e x)^{3/2} \sqrt [4]{12-3 e^2 x^2}} \, dx=\int { \frac {1}{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}} {\left (e x + 2\right )}^{\frac {3}{2}}} \,d x } \]
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Exception generated. \[ \int \frac {1}{(2+e x)^{3/2} \sqrt [4]{12-3 e^2 x^2}} \, dx=\text {Exception raised: TypeError} \]
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Time = 10.47 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.69 \[ \int \frac {1}{(2+e x)^{3/2} \sqrt [4]{12-3 e^2 x^2}} \, dx=-\frac {{\left (12-3\,e^2\,x^2\right )}^{3/4}}{9\,e\,{\left (e\,x+2\right )}^{3/2}} \]
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