Optimal. Leaf size=71 \[ -\frac{\left (4-e^2 x^2\right )^{3/4}}{21 \sqrt [4]{3} e (e x+2)^{3/2}}-\frac{\left (4-e^2 x^2\right )^{3/4}}{7 \sqrt [4]{3} e (e x+2)^{5/2}} \]
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Rubi [A] time = 0.0264299, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {659, 651} \[ -\frac{\left (4-e^2 x^2\right )^{3/4}}{21 \sqrt [4]{3} e (e x+2)^{3/2}}-\frac{\left (4-e^2 x^2\right )^{3/4}}{7 \sqrt [4]{3} e (e x+2)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 659
Rule 651
Rubi steps
\begin{align*} \int \frac{1}{(2+e x)^{5/2} \sqrt [4]{12-3 e^2 x^2}} \, dx &=-\frac{\left (4-e^2 x^2\right )^{3/4}}{7 \sqrt [4]{3} e (2+e x)^{5/2}}+\frac{1}{7} \int \frac{1}{(2+e x)^{3/2} \sqrt [4]{12-3 e^2 x^2}} \, dx\\ &=-\frac{\left (4-e^2 x^2\right )^{3/4}}{7 \sqrt [4]{3} e (2+e x)^{5/2}}-\frac{\left (4-e^2 x^2\right )^{3/4}}{21 \sqrt [4]{3} e (2+e x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0572088, size = 40, normalized size = 0.56 \[ \frac{(e x-2) (e x+5)}{21 e (e x+2)^{3/2} \sqrt [4]{12-3 e^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 35, normalized size = 0.5 \begin{align*}{\frac{ \left ( ex-2 \right ) \left ( ex+5 \right ) }{21\,e} \left ( ex+2 \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt [4]{-3\,{e}^{2}{x}^{2}+12}}}} \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{1}{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}}{\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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Fricas [A] time = 1.81542, size = 127, normalized size = 1.79 \begin{align*} -\frac{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{3}{4}}{\left (e x + 5\right )} \sqrt{e x + 2}}{63 \,{\left (e^{4} x^{3} + 6 \, e^{3} x^{2} + 12 \, e^{2} x + 8 \, 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{1}{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}}{\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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