Optimal. Leaf size=85 \[ \frac{2 (2-e x)^{7/2}}{7 \sqrt{3} e}-\frac{8 \sqrt{3} (2-e x)^{5/2}}{5 e}+\frac{32 (2-e x)^{3/2}}{\sqrt{3} e}-\frac{128 \sqrt{2-e x}}{\sqrt{3} e} \]
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Rubi [A] time = 0.0233863, antiderivative size = 85, 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} \[ \frac{2 (2-e x)^{7/2}}{7 \sqrt{3} e}-\frac{8 \sqrt{3} (2-e x)^{5/2}}{5 e}+\frac{32 (2-e x)^{3/2}}{\sqrt{3} e}-\frac{128 \sqrt{2-e x}}{\sqrt{3} e} \]
Antiderivative was successfully verified.
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Rule 627
Rule 43
Rubi steps
\begin{align*} \int \frac{(2+e x)^{7/2}}{\sqrt{12-3 e^2 x^2}} \, dx &=\int \frac{(2+e x)^3}{\sqrt{6-3 e x}} \, dx\\ &=\int \left (\frac{64}{\sqrt{6-3 e x}}-16 \sqrt{6-3 e x}+\frac{4}{3} (6-3 e x)^{3/2}-\frac{1}{27} (6-3 e x)^{5/2}\right ) \, dx\\ &=-\frac{128 \sqrt{2-e x}}{\sqrt{3} e}+\frac{32 (2-e x)^{3/2}}{\sqrt{3} e}-\frac{8 \sqrt{3} (2-e x)^{5/2}}{5 e}+\frac{2 (2-e x)^{7/2}}{7 \sqrt{3} e}\\ \end{align*}
Mathematica [A] time = 0.0827437, size = 57, normalized size = 0.67 \[ \frac{2 (e x-2) \sqrt{e x+2} \left (5 e^3 x^3+54 e^2 x^2+284 e x+1416\right )}{35 e \sqrt{12-3 e^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 52, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,ex-4 \right ) \left ( 5\,{e}^{3}{x}^{3}+54\,{e}^{2}{x}^{2}+284\,ex+1416 \right ) }{35\,e}\sqrt{ex+2}{\frac{1}{\sqrt{-3\,{e}^{2}{x}^{2}+12}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.90065, size = 61, normalized size = 0.72 \begin{align*} -\frac{2 i \, \sqrt{3}{\left (5 \, e^{4} x^{4} + 44 \, e^{3} x^{3} + 176 \, e^{2} x^{2} + 848 \, e x - 2832\right )}}{105 \, \sqrt{e x - 2} e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81925, size = 135, normalized size = 1.59 \begin{align*} -\frac{2 \,{\left (5 \, e^{3} x^{3} + 54 \, e^{2} x^{2} + 284 \, e x + 1416\right )} \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2}}{105 \,{\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 \frac{{\left (e x + 2\right )}^{\frac{7}{2}}}{\sqrt{-3 \, e^{2} x^{2} + 12}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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