Optimal. Leaf size=109 \[ -\frac {6 \sqrt {3} (2-e x)^{13/2}}{13 e}+\frac {96 \sqrt {3} (2-e x)^{11/2}}{11 e}-\frac {64 \sqrt {3} (2-e x)^{9/2}}{e}+\frac {1536 \sqrt {3} (2-e x)^{7/2}}{7 e}-\frac {1536 \sqrt {3} (2-e x)^{5/2}}{5 e} \]
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Rubi [A] time = 0.03, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {627, 43} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 627
Rubi steps
\begin {align*} \int (2+e x)^{5/2} \left (12-3 e^2 x^2\right )^{3/2} \, dx &=\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*}
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Mathematica [A] time = 0.07, size = 67, normalized size = 0.61 \begin {gather*} -\frac {2 (e x-2)^2 \sqrt {12-3 e^2 x^2} \left (1155 e^4 x^4+12600 e^3 x^3+56840 e^2 x^2+133600 e x+154928\right )}{5005 e \sqrt {e x+2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.31, size = 99, normalized size = 0.91 \begin {gather*} -\frac {2 \left (4 (e x+2)-(e x+2)^2\right )^{5/2} \left (1155 \sqrt {3} (e x+2)^4+3360 \sqrt {3} (e x+2)^3+8960 \sqrt {3} (e x+2)^2+20480 \sqrt {3} (e x+2)+32768 \sqrt {3}\right )}{5005 e (e x+2)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 78, normalized size = 0.72 \begin {gather*} -\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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 60, normalized size = 0.55 \begin {gather*} \frac {2 \left (e x -2\right ) \left (1155 e^{4} x^{4}+12600 e^{3} x^{3}+56840 e^{2} x^{2}+133600 e x +154928\right ) \left (-3 e^{2} x^{2}+12\right )^{\frac {3}{2}}}{15015 \left (e x +2\right )^{\frac {3}{2}} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 3.09, size = 93, normalized size = 0.85 \begin {gather*} -\frac {{\left (2310 i \, \sqrt {3} e^{6} x^{6} + 15960 i \, \sqrt {3} e^{5} x^{5} + 22120 i \, \sqrt {3} e^{4} x^{4} - 86720 i \, \sqrt {3} e^{3} x^{3} - 304224 i \, \sqrt {3} e^{2} x^{2} - 170624 i \, \sqrt {3} e x + 1239424 i \, \sqrt {3}\right )} {\left (e x + 2\right )} \sqrt {e x - 2}}{5005 \, {\left (e^{2} x + 2 \, e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.27, size = 87, normalized size = 0.80 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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