Integrand size = 24, antiderivative size = 109 \[ \int (2+e x)^{5/2} \left (12-3 e^2 x^2\right )^{3/2} \, dx=-\frac {1536 \sqrt {3} (2-e x)^{5/2}}{5 e}+\frac {1536 \sqrt {3} (2-e x)^{7/2}}{7 e}-\frac {64 \sqrt {3} (2-e x)^{9/2}}{e}+\frac {96 \sqrt {3} (2-e x)^{11/2}}{11 e}-\frac {6 \sqrt {3} (2-e x)^{13/2}}{13 e} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 109, 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} \[ \int (2+e x)^{5/2} \left (12-3 e^2 x^2\right )^{3/2} \, dx=-\frac {6 \sqrt {3} (2-e x)^{13/2}}{13 e}+\frac {96 \sqrt {3} (2-e x)^{11/2}}{11 e}-\frac {64 \sqrt {3} (2-e x)^{9/2}}{e}+\frac {1536 \sqrt {3} (2-e x)^{7/2}}{7 e}-\frac {1536 \sqrt {3} (2-e x)^{5/2}}{5 e} \]
[In]
[Out]
Rule 45
Rule 641
Rubi steps \begin{align*} \text {integral}& = \int (6-3 e x)^{3/2} (2+e x)^4 \, dx \\ & = \int \left (256 (6-3 e x)^{3/2}-\frac {256}{3} (6-3 e x)^{5/2}+\frac {32}{3} (6-3 e x)^{7/2}-\frac {16}{27} (6-3 e x)^{9/2}+\frac {1}{81} (6-3 e x)^{11/2}\right ) \, dx \\ & = -\frac {1536 \sqrt {3} (2-e x)^{5/2}}{5 e}+\frac {1536 \sqrt {3} (2-e x)^{7/2}}{7 e}-\frac {64 \sqrt {3} (2-e x)^{9/2}}{e}+\frac {96 \sqrt {3} (2-e x)^{11/2}}{11 e}-\frac {6 \sqrt {3} (2-e x)^{13/2}}{13 e} \\ \end{align*}
Time = 0.40 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.61 \[ \int (2+e x)^{5/2} \left (12-3 e^2 x^2\right )^{3/2} \, dx=-\frac {2 (-2+e x)^2 \sqrt {12-3 e^2 x^2} \left (154928+133600 e x+56840 e^2 x^2+12600 e^3 x^3+1155 e^4 x^4\right )}{5005 e \sqrt {2+e x}} \]
[In]
[Out]
Time = 2.26 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.55
method | result | size |
gosper | \(\frac {2 \left (e x -2\right ) \left (1155 e^{4} x^{4}+12600 e^{3} x^{3}+56840 x^{2} e^{2}+133600 e x +154928\right ) \left (-3 x^{2} e^{2}+12\right )^{\frac {3}{2}}}{15015 e \left (e x +2\right )^{\frac {3}{2}}}\) | \(60\) |
default | \(-\frac {2 \sqrt {-3 x^{2} e^{2}+12}\, \left (e x -2\right )^{2} \left (1155 e^{4} x^{4}+12600 e^{3} x^{3}+56840 x^{2} e^{2}+133600 e x +154928\right )}{5005 \sqrt {e x +2}\, e}\) | \(62\) |
risch | \(\frac {6 \sqrt {\frac {-3 x^{2} e^{2}+12}{e x +2}}\, \sqrt {e x +2}\, \left (1155 e^{6} x^{6}+7980 e^{5} x^{5}+11060 e^{4} x^{4}-43360 e^{3} x^{3}-152112 x^{2} e^{2}-85312 e x +619712\right ) \left (e x -2\right )}{5005 \sqrt {-3 x^{2} e^{2}+12}\, e \sqrt {-3 e x +6}}\) | \(104\) |
[In]
[Out]
none
Time = 0.38 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.72 \[ \int (2+e x)^{5/2} \left (12-3 e^2 x^2\right )^{3/2} \, dx=-\frac {2 \, {\left (1155 \, e^{6} x^{6} + 7980 \, e^{5} x^{5} + 11060 \, e^{4} x^{4} - 43360 \, e^{3} x^{3} - 152112 \, e^{2} x^{2} - 85312 \, e x + 619712\right )} \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2}}{5005 \, {\left (e^{2} x + 2 \, e\right )}} \]
[In]
[Out]
Timed out. \[ \int (2+e x)^{5/2} \left (12-3 e^2 x^2\right )^{3/2} \, dx=\text {Timed out} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.85 \[ \int (2+e x)^{5/2} \left (12-3 e^2 x^2\right )^{3/2} \, dx=\frac {2 \, {\left (-1155 i \, \sqrt {3} e^{6} x^{6} - 7980 i \, \sqrt {3} e^{5} x^{5} - 11060 i \, \sqrt {3} e^{4} x^{4} + 43360 i \, \sqrt {3} e^{3} x^{3} + 152112 i \, \sqrt {3} e^{2} x^{2} + 85312 i \, \sqrt {3} e x - 619712 i \, \sqrt {3}\right )} {\left (e x + 2\right )} \sqrt {e x - 2}}{5005 \, {\left (e^{2} x + 2 \, e\right )}} \]
[In]
[Out]
Exception generated. \[ \int (2+e x)^{5/2} \left (12-3 e^2 x^2\right )^{3/2} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Time = 0.26 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.80 \[ \int (2+e x)^{5/2} \left (12-3 e^2 x^2\right )^{3/2} \, dx=\frac {2\,\sqrt {12-3\,e^2\,x^2}\,\sqrt {e\,x+2}\,\left (-1155\,e^5\,x^5-5670\,e^4\,x^4+280\,e^3\,x^3+42800\,e^2\,x^2+66512\,e\,x-47712\right )}{5005\,e}-\frac {1048576\,\sqrt {12-3\,e^2\,x^2}}{5005\,e\,\sqrt {e\,x+2}} \]
[In]
[Out]