Optimal. Leaf size=113 \[ -\frac{\sqrt{3} (2-e x)^{3/2}}{e (e x+2)^3}-\frac{3 \sqrt{3} \sqrt{2-e x}}{32 e (e x+2)}+\frac{3 \sqrt{3} \sqrt{2-e x}}{4 e (e x+2)^2}-\frac{3 \sqrt{3} \tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{64 e} \]
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Rubi [A] time = 0.0389677, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {627, 47, 51, 63, 206} \[ -\frac{\sqrt{3} (2-e x)^{3/2}}{e (e x+2)^3}-\frac{3 \sqrt{3} \sqrt{2-e x}}{32 e (e x+2)}+\frac{3 \sqrt{3} \sqrt{2-e x}}{4 e (e x+2)^2}-\frac{3 \sqrt{3} \tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{64 e} \]
Antiderivative was successfully verified.
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Rule 627
Rule 47
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (12-3 e^2 x^2\right )^{3/2}}{(2+e x)^{11/2}} \, dx &=\int \frac{(6-3 e x)^{3/2}}{(2+e x)^4} \, dx\\ &=-\frac{\sqrt{3} (2-e x)^{3/2}}{e (2+e x)^3}-\frac{3}{2} \int \frac{\sqrt{6-3 e x}}{(2+e x)^3} \, dx\\ &=-\frac{\sqrt{3} (2-e x)^{3/2}}{e (2+e x)^3}+\frac{3 \sqrt{3} \sqrt{2-e x}}{4 e (2+e x)^2}+\frac{9}{8} \int \frac{1}{\sqrt{6-3 e x} (2+e x)^2} \, dx\\ &=-\frac{\sqrt{3} (2-e x)^{3/2}}{e (2+e x)^3}+\frac{3 \sqrt{3} \sqrt{2-e x}}{4 e (2+e x)^2}-\frac{3 \sqrt{3} \sqrt{2-e x}}{32 e (2+e x)}+\frac{9}{64} \int \frac{1}{\sqrt{6-3 e x} (2+e x)} \, dx\\ &=-\frac{\sqrt{3} (2-e x)^{3/2}}{e (2+e x)^3}+\frac{3 \sqrt{3} \sqrt{2-e x}}{4 e (2+e x)^2}-\frac{3 \sqrt{3} \sqrt{2-e x}}{32 e (2+e x)}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{4-\frac{x^2}{3}} \, dx,x,\sqrt{6-3 e x}\right )}{32 e}\\ &=-\frac{\sqrt{3} (2-e x)^{3/2}}{e (2+e x)^3}+\frac{3 \sqrt{3} \sqrt{2-e x}}{4 e (2+e x)^2}-\frac{3 \sqrt{3} \sqrt{2-e x}}{32 e (2+e x)}-\frac{3 \sqrt{3} \tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{64 e}\\ \end{align*}
Mathematica [C] time = 0.0846051, size = 55, normalized size = 0.49 \[ -\frac{3 (e x-2)^2 \sqrt{12-3 e^2 x^2} \, _2F_1\left (\frac{5}{2},4;\frac{7}{2};\frac{1}{2}-\frac{e x}{4}\right )}{640 e \sqrt{e x+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.154, size = 167, normalized size = 1.5 \begin{align*} -{\frac{\sqrt{3}}{64\,e}\sqrt{-{e}^{2}{x}^{2}+4} \left ( 3\,\sqrt{3}{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ){x}^{3}{e}^{3}+18\,\sqrt{3}{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ){x}^{2}{e}^{2}+6\,{x}^{2}{e}^{2}\sqrt{-3\,ex+6}+36\,\sqrt{3}{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ) xe-88\,xe\sqrt{-3\,ex+6}+24\,\sqrt{3}{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ) +56\,\sqrt{-3\,ex+6} \right ){\frac{1}{\sqrt{ \left ( ex+2 \right ) ^{7}}}}{\frac{1}{\sqrt{-3\,ex+6}}}} \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{11}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.85877, size = 390, normalized size = 3.45 \begin{align*} \frac{3 \, \sqrt{3}{\left (e^{4} x^{4} + 8 \, e^{3} x^{3} + 24 \, e^{2} x^{2} + 32 \, e x + 16\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 ) - 4 \,{\left (3 \, e^{2} x^{2} - 44 \, e x + 28\right )} \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2}}{128 \,{\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 [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{11}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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