Optimal. Leaf size=106 \[ -\frac{2 \left (4-e^2 x^2\right )^{5/4}}{195\ 3^{3/4} e (e x+2)^{5/2}}-\frac{2 \left (4-e^2 x^2\right )^{5/4}}{39\ 3^{3/4} e (e x+2)^{7/2}}-\frac{\sqrt [4]{3} \left (4-e^2 x^2\right )^{5/4}}{13 e (e x+2)^{9/2}} \]
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Rubi [A] time = 0.040833, antiderivative size = 106, 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 = {659, 651} \[ -\frac{2 \left (4-e^2 x^2\right )^{5/4}}{195\ 3^{3/4} e (e x+2)^{5/2}}-\frac{2 \left (4-e^2 x^2\right )^{5/4}}{39\ 3^{3/4} e (e x+2)^{7/2}}-\frac{\sqrt [4]{3} \left (4-e^2 x^2\right )^{5/4}}{13 e (e x+2)^{9/2}} \]
Antiderivative was successfully verified.
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Rule 659
Rule 651
Rubi steps
\begin{align*} \int \frac{\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{9/2}} \, dx &=-\frac{\sqrt [4]{3} \left (4-e^2 x^2\right )^{5/4}}{13 e (2+e x)^{9/2}}+\frac{2}{13} \int \frac{\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{7/2}} \, dx\\ &=-\frac{\sqrt [4]{3} \left (4-e^2 x^2\right )^{5/4}}{13 e (2+e x)^{9/2}}-\frac{2 \left (4-e^2 x^2\right )^{5/4}}{39\ 3^{3/4} e (2+e x)^{7/2}}+\frac{2}{117} \int \frac{\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{5/2}} \, dx\\ &=-\frac{\sqrt [4]{3} \left (4-e^2 x^2\right )^{5/4}}{13 e (2+e x)^{9/2}}-\frac{2 \left (4-e^2 x^2\right )^{5/4}}{39\ 3^{3/4} e (2+e x)^{7/2}}-\frac{2 \left (4-e^2 x^2\right )^{5/4}}{195\ 3^{3/4} e (2+e x)^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0762167, size = 57, normalized size = 0.54 \[ \frac{\sqrt [4]{4-e^2 x^2} \left (2 e^3 x^3+14 e^2 x^2+37 e x-146\right )}{195\ 3^{3/4} e (e x+2)^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 44, normalized size = 0.4 \begin{align*}{\frac{ \left ( ex-2 \right ) \left ( 2\,{e}^{2}{x}^{2}+18\,ex+73 \right ) }{585\,e}\sqrt [4]{-3\,{e}^{2}{x}^{2}+12} \left ( ex+2 \right ) ^{-{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}}}{{\left (e x + 2\right )}^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.84171, size = 186, normalized size = 1.75 \begin{align*} \frac{{\left (2 \, e^{3} x^{3} + 14 \, e^{2} x^{2} + 37 \, e x - 146\right )}{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}} \sqrt{e x + 2}}{585 \,{\left (e^{5} x^{4} + 8 \, e^{4} x^{3} + 24 \, e^{3} x^{2} + 32 \, e^{2} x + 16 \, 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 [A] time = 1.32091, size = 197, normalized size = 1.86 \begin{align*} -\frac{1}{9360} \cdot 3^{\frac{1}{4}}{\left (\frac{117 \,{\left (-{\left (x e + 2\right )}^{2} + 4 \, x e + 8\right )}^{\frac{1}{4}}{\left (\frac{4}{x e + 2} - 1\right )}}{\sqrt{x e + 2}} + \frac{130 \,{\left ({\left (x e + 2\right )}^{2} - 8 \, x e\right )}{\left (-{\left (x e + 2\right )}^{2} + 4 \, x e + 8\right )}^{\frac{1}{4}}}{{\left (x e + 2\right )}^{\frac{5}{2}}} - \frac{45 \,{\left ({\left (x e + 2\right )}^{3} - 12 \,{\left (x e + 2\right )}^{2} + 48 \, x e + 32\right )}{\left (-{\left (x e + 2\right )}^{2} + 4 \, x e + 8\right )}^{\frac{1}{4}}}{{\left (x e + 2\right )}^{\frac{7}{2}}}\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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