Integrand size = 24, antiderivative size = 111 \[ \int \frac {(2+e x)^{11/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=\frac {512}{3 \sqrt {3} e \sqrt {2-e x}}+\frac {512 \sqrt {2-e x}}{3 \sqrt {3} e}-\frac {64 (2-e x)^{3/2}}{3 \sqrt {3} e}+\frac {32 (2-e x)^{5/2}}{15 \sqrt {3} e}-\frac {2 (2-e x)^{7/2}}{21 \sqrt {3} e} \]
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Time = 0.02 (sec) , antiderivative size = 111, 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} \[ \int \frac {(2+e x)^{11/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=-\frac {2 (2-e x)^{7/2}}{21 \sqrt {3} e}+\frac {32 (2-e x)^{5/2}}{15 \sqrt {3} e}-\frac {64 (2-e x)^{3/2}}{3 \sqrt {3} e}+\frac {512 \sqrt {2-e x}}{3 \sqrt {3} e}+\frac {512}{3 \sqrt {3} e \sqrt {2-e x}} \]
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Rule 45
Rule 641
Rubi steps \begin{align*} \text {integral}& = \int \frac {(2+e x)^4}{(6-3 e x)^{3/2}} \, dx \\ & = \int \left (\frac {256}{(6-3 e x)^{3/2}}-\frac {256}{3 \sqrt {6-3 e x}}+\frac {32}{3} \sqrt {6-3 e x}-\frac {16}{27} (6-3 e x)^{3/2}+\frac {1}{81} (6-3 e x)^{5/2}\right ) \, dx \\ & = \frac {512}{3 \sqrt {3} e \sqrt {2-e x}}+\frac {512 \sqrt {2-e x}}{3 \sqrt {3} e}-\frac {64 (2-e x)^{3/2}}{3 \sqrt {3} e}+\frac {32 (2-e x)^{5/2}}{15 \sqrt {3} e}-\frac {2 (2-e x)^{7/2}}{21 \sqrt {3} e} \\ \end{align*}
Time = 0.56 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.61 \[ \int \frac {(2+e x)^{11/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=\frac {2 \sqrt {4-e^2 x^2} \left (-23216+5664 e x+568 e^2 x^2+72 e^3 x^3+5 e^4 x^4\right )}{105 e (-2+e x) \sqrt {6+3 e x}} \]
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Time = 2.31 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.54
method | result | size |
gosper | \(\frac {2 \left (e x -2\right ) \left (5 e^{4} x^{4}+72 e^{3} x^{3}+568 x^{2} e^{2}+5664 e x -23216\right ) \left (e x +2\right )^{\frac {3}{2}}}{35 e \left (-3 x^{2} e^{2}+12\right )^{\frac {3}{2}}}\) | \(60\) |
default | \(\frac {2 \sqrt {-3 x^{2} e^{2}+12}\, \left (5 e^{4} x^{4}+72 e^{3} x^{3}+568 x^{2} e^{2}+5664 e x -23216\right )}{315 \sqrt {e x +2}\, \left (e x -2\right ) e}\) | \(62\) |
risch | \(-\frac {2 \left (5 e^{3} x^{3}+82 x^{2} e^{2}+732 e x +7128\right ) \left (e x -2\right ) \sqrt {\frac {-3 x^{2} e^{2}+12}{e x +2}}\, \sqrt {e x +2}}{105 e \sqrt {-3 e x +6}\, \sqrt {-3 x^{2} e^{2}+12}}+\frac {512 \sqrt {\frac {-3 x^{2} e^{2}+12}{e x +2}}\, \sqrt {e x +2}}{3 e \sqrt {-3 e x +6}\, \sqrt {-3 x^{2} e^{2}+12}}\) | \(133\) |
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Time = 0.28 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.58 \[ \int \frac {(2+e x)^{11/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=\frac {2 \, {\left (5 \, e^{4} x^{4} + 72 \, e^{3} x^{3} + 568 \, e^{2} x^{2} + 5664 \, e x - 23216\right )} \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2}}{315 \, {\left (e^{3} x^{2} - 4 \, e\right )}} \]
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Timed out. \[ \int \frac {(2+e x)^{11/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=\text {Timed out} \]
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Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.52 \[ \int \frac {(2+e x)^{11/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=-\frac {2 \, {\left (-5 i \, \sqrt {3} e^{4} x^{4} - 72 i \, \sqrt {3} e^{3} x^{3} - 568 i \, \sqrt {3} e^{2} x^{2} - 5664 i \, \sqrt {3} e x + 23216 i \, \sqrt {3}\right )}}{315 \, \sqrt {e x - 2} e} \]
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Exception generated. \[ \int \frac {(2+e x)^{11/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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Time = 0.30 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.84 \[ \int \frac {(2+e x)^{11/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx=-\frac {\sqrt {12-3\,e^2\,x^2}\,\left (\frac {16\,x^3\,\sqrt {e\,x+2}}{35}-\frac {46432\,\sqrt {e\,x+2}}{315\,e^3}+\frac {3776\,x\,\sqrt {e\,x+2}}{105\,e^2}+\frac {2\,e\,x^4\,\sqrt {e\,x+2}}{63}+\frac {1136\,x^2\,\sqrt {e\,x+2}}{315\,e}\right )}{\frac {4}{e^2}-x^2} \]
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