Optimal. Leaf size=87 \[ \frac {6 \sqrt {3} (2-e x)^{11/2}}{11 e}-\frac {8 \sqrt {3} (2-e x)^{9/2}}{e}+\frac {288 \sqrt {3} (2-e x)^{7/2}}{7 e}-\frac {384 \sqrt {3} (2-e x)^{5/2}}{5 e} \]
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Rubi [A] time = 0.02, antiderivative size = 87, 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)^{11/2}}{11 e}-\frac {8 \sqrt {3} (2-e x)^{9/2}}{e}+\frac {288 \sqrt {3} (2-e x)^{7/2}}{7 e}-\frac {384 \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)^{3/2} \left (12-3 e^2 x^2\right )^{3/2} \, dx &=\int (6-3 e x)^{3/2} (2+e x)^3 \, dx\\ &=\int \left (64 (6-3 e x)^{3/2}-16 (6-3 e x)^{5/2}+\frac {4}{3} (6-3 e x)^{7/2}-\frac {1}{27} (6-3 e x)^{9/2}\right ) \, dx\\ &=-\frac {384 \sqrt {3} (2-e x)^{5/2}}{5 e}+\frac {288 \sqrt {3} (2-e x)^{7/2}}{7 e}-\frac {8 \sqrt {3} (2-e x)^{9/2}}{e}+\frac {6 \sqrt {3} (2-e x)^{11/2}}{11 e}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 59, normalized size = 0.68 \begin {gather*} -\frac {2 (e x-2)^2 \sqrt {12-3 e^2 x^2} \left (105 e^3 x^3+910 e^2 x^2+3020 e x+4264\right )}{385 e \sqrt {e x+2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.29, size = 85, normalized size = 0.98 \begin {gather*} -\frac {2 \left (4 (e x+2)-(e x+2)^2\right )^{5/2} \left (105 \sqrt {3} (e x+2)^3+280 \sqrt {3} (e x+2)^2+640 \sqrt {3} (e x+2)+1024 \sqrt {3}\right )}{385 e (e x+2)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 70, normalized size = 0.80 \begin {gather*} -\frac {2 \, {\left (105 \, e^{5} x^{5} + 490 \, e^{4} x^{4} - 200 \, e^{3} x^{3} - 4176 \, e^{2} x^{2} - 4976 \, e x + 17056\right )} \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2}}{385 \, {\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] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {3}{2}} {\left (e x + 2\right )}^{\frac {3}{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 52, normalized size = 0.60 \begin {gather*} \frac {2 \left (e x -2\right ) \left (105 e^{3} x^{3}+910 e^{2} x^{2}+3020 e x +4264\right ) \left (-3 e^{2} x^{2}+12\right )^{\frac {3}{2}}}{1155 \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 = 3.19, size = 82, normalized size = 0.94 \begin {gather*} -\frac {{\left (210 i \, \sqrt {3} e^{5} x^{5} + 980 i \, \sqrt {3} e^{4} x^{4} - 400 i \, \sqrt {3} e^{3} x^{3} - 8352 i \, \sqrt {3} e^{2} x^{2} - 9952 i \, \sqrt {3} e x + 34112 i \, \sqrt {3}\right )} {\left (e x + 2\right )} \sqrt {e x - 2}}{385 \, {\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 = 0.59, size = 53, normalized size = 0.61 \begin {gather*} -\frac {2\,\sqrt {12-3\,e^2\,x^2}\,{\left (e\,x-2\right )}^2\,\left (105\,e^3\,x^3+910\,e^2\,x^2+3020\,e\,x+4264\right )}{385\,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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