Optimal. Leaf size=73 \[ -\frac{3 \sqrt{3} (2-e x)^{3/2}}{e (e x+2)}-\frac{9 \sqrt{3} \sqrt{2-e x}}{e}+\frac{18 \sqrt{3} \tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{e} \]
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Rubi [A] time = 0.0276186, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {627, 47, 50, 63, 206} \[ -\frac{3 \sqrt{3} (2-e x)^{3/2}}{e (e x+2)}-\frac{9 \sqrt{3} \sqrt{2-e x}}{e}+\frac{18 \sqrt{3} \tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{e} \]
Antiderivative was successfully verified.
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Rule 627
Rule 47
Rule 50
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (12-3 e^2 x^2\right )^{3/2}}{(2+e x)^{7/2}} \, dx &=\int \frac{(6-3 e x)^{3/2}}{(2+e x)^2} \, dx\\ &=-\frac{3 \sqrt{3} (2-e x)^{3/2}}{e (2+e x)}-\frac{9}{2} \int \frac{\sqrt{6-3 e x}}{2+e x} \, dx\\ &=-\frac{9 \sqrt{3} \sqrt{2-e x}}{e}-\frac{3 \sqrt{3} (2-e x)^{3/2}}{e (2+e x)}-54 \int \frac{1}{\sqrt{6-3 e x} (2+e x)} \, dx\\ &=-\frac{9 \sqrt{3} \sqrt{2-e x}}{e}-\frac{3 \sqrt{3} (2-e x)^{3/2}}{e (2+e x)}+\frac{36 \operatorname{Subst}\left (\int \frac{1}{4-\frac{x^2}{3}} \, dx,x,\sqrt{6-3 e x}\right )}{e}\\ &=-\frac{9 \sqrt{3} \sqrt{2-e x}}{e}-\frac{3 \sqrt{3} (2-e x)^{3/2}}{e (2+e x)}+\frac{18 \sqrt{3} \tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{e}\\ \end{align*}
Mathematica [C] time = 0.0632678, size = 55, normalized size = 0.75 \[ -\frac{3 (e x-2)^2 \sqrt{12-3 e^2 x^2} \, _2F_1\left (2,\frac{5}{2};\frac{7}{2};\frac{1}{2}-\frac{e x}{4}\right )}{40 e \sqrt{e x+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.131, size = 101, normalized size = 1.4 \begin{align*} 6\,{\frac{\sqrt{-{e}^{2}{x}^{2}+4} \left ( 3\,\sqrt{3}{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ) xe-xe\sqrt{-3\,ex+6}+6\,\sqrt{3}{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ) -4\,\sqrt{-3\,ex+6} \right ) \sqrt{3}}{\sqrt{ \left ( ex+2 \right ) ^{3}}\sqrt{-3\,ex+6}e}} \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{3}{2}}}{{\left (e x + 2\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.81619, size = 290, normalized size = 3.97 \begin{align*} \frac{3 \,{\left (3 \, \sqrt{3}{\left (e^{2} x^{2} + 4 \, e x + 4\right )} \log \left (-\frac{3 \, e^{2} x^{2} - 12 \, e x - 4 \, \sqrt{3} \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2} - 36}{e^{2} x^{2} + 4 \, e x + 4}\right ) - 2 \, \sqrt{-3 \, e^{2} x^{2} + 12}{\left (e x + 4\right )} \sqrt{e x + 2}\right )}}{e^{3} x^{2} + 4 \, e^{2} x + 4 \, e} \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 (-3 \, e^{2} x^{2} + 12\right )}^{\frac{3}{2}}}{{\left (e x + 2\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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