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 = 0.0274405, antiderivative size = 109, normalized size of antiderivative = 1., 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} \[ -\frac{6 \sqrt{3} (2-e x)^{13/2}}{13 e}+\frac{96 \sqrt{3} (2-e x)^{11/2}}{11 e}-\frac{64 \sqrt{3} (2-e x)^{9/2}}{e}+\frac{1536 \sqrt{3} (2-e x)^{7/2}}{7 e}-\frac{1536 \sqrt{3} (2-e x)^{5/2}}{5 e} \]
Antiderivative was successfully verified.
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Rule 627
Rule 43
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*}
Mathematica [A] time = 0.0682521, size = 67, normalized size = 0.61 \[ -\frac{2 (e x-2)^2 \sqrt{12-3 e^2 x^2} \left (1155 e^4 x^4+12600 e^3 x^3+56840 e^2 x^2+133600 e x+154928\right )}{5005 e \sqrt{e x+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 60, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,ex-4 \right ) \left ( 1155\,{e}^{4}{x}^{4}+12600\,{e}^{3}{x}^{3}+56840\,{e}^{2}{x}^{2}+133600\,ex+154928 \right ) }{15015\,e} \left ( -3\,{e}^{2}{x}^{2}+12 \right ) ^{{\frac{3}{2}}} \left ( ex+2 \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.77502, size = 126, normalized size = 1.16 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.84497, size = 215, normalized size = 1.97 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{3}{2}}{\left (e x + 2\right )}^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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