Optimal. Leaf size=108 \[ \frac{5}{256 \sqrt{3} e \sqrt{2-e x}}-\frac{5}{192 \sqrt{3} e \sqrt{2-e x} (e x+2)}-\frac{1}{24 \sqrt{3} e \sqrt{2-e x} (e x+2)^2}-\frac{5 \tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{512 \sqrt{3} e} \]
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Rubi [A] time = 0.0414894, antiderivative size = 115, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {627, 51, 63, 206} \[ -\frac{5 \sqrt{2-e x}}{256 \sqrt{3} e (e x+2)}-\frac{5 \sqrt{2-e x}}{96 \sqrt{3} e (e x+2)^2}+\frac{1}{6 \sqrt{3} e \sqrt{2-e x} (e x+2)^2}-\frac{5 \tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{512 \sqrt{3} e} \]
Antiderivative was successfully verified.
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Rule 627
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{(2+e x)^{3/2} \left (12-3 e^2 x^2\right )^{3/2}} \, dx &=\int \frac{1}{(6-3 e x)^{3/2} (2+e x)^3} \, dx\\ &=\frac{1}{6 \sqrt{3} e \sqrt{2-e x} (2+e x)^2}+\frac{5}{12} \int \frac{1}{\sqrt{6-3 e x} (2+e x)^3} \, dx\\ &=\frac{1}{6 \sqrt{3} e \sqrt{2-e x} (2+e x)^2}-\frac{5 \sqrt{2-e x}}{96 \sqrt{3} e (2+e x)^2}+\frac{5}{64} \int \frac{1}{\sqrt{6-3 e x} (2+e x)^2} \, dx\\ &=\frac{1}{6 \sqrt{3} e \sqrt{2-e x} (2+e x)^2}-\frac{5 \sqrt{2-e x}}{96 \sqrt{3} e (2+e x)^2}-\frac{5 \sqrt{2-e x}}{256 \sqrt{3} e (2+e x)}+\frac{5}{512} \int \frac{1}{\sqrt{6-3 e x} (2+e x)} \, dx\\ &=\frac{1}{6 \sqrt{3} e \sqrt{2-e x} (2+e x)^2}-\frac{5 \sqrt{2-e x}}{96 \sqrt{3} e (2+e x)^2}-\frac{5 \sqrt{2-e x}}{256 \sqrt{3} e (2+e x)}-\frac{5 \operatorname{Subst}\left (\int \frac{1}{4-\frac{x^2}{3}} \, dx,x,\sqrt{6-3 e x}\right )}{768 e}\\ &=\frac{1}{6 \sqrt{3} e \sqrt{2-e x} (2+e x)^2}-\frac{5 \sqrt{2-e x}}{96 \sqrt{3} e (2+e x)^2}-\frac{5 \sqrt{2-e x}}{256 \sqrt{3} e (2+e x)}-\frac{5 \tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{512 \sqrt{3} e}\\ \end{align*}
Mathematica [C] time = 0.0501724, size = 48, normalized size = 0.44 \[ \frac{\sqrt{e x+2} \, _2F_1\left (-\frac{1}{2},3;\frac{1}{2};\frac{1}{2}-\frac{e x}{4}\right )}{96 e \sqrt{12-3 e^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 135, normalized size = 1.3 \begin{align*}{\frac{1}{ \left ( 4608\,ex-9216 \right ) e}\sqrt{-3\,{e}^{2}{x}^{2}+12} \left ( 5\,\sqrt{3}{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ) \sqrt{-3\,ex+6}{x}^{2}{e}^{2}+20\,\sqrt{3}{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ) \sqrt{-3\,ex+6}xe-30\,{e}^{2}{x}^{2}+20\,\sqrt{3}{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ) \sqrt{-3\,ex+6}-80\,ex+24 \right ) \left ( ex+2 \right ) ^{-{\frac{5}{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{1}{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{3}{2}}{\left (e x + 2\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.83962, size = 359, normalized size = 3.32 \begin{align*} \frac{15 \, \sqrt{3}{\left (e^{4} x^{4} + 4 \, e^{3} x^{3} - 16 \, 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 (15 \, e^{2} x^{2} + 40 \, e x - 12\right )} \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2}}{9216 \,{\left (e^{5} x^{4} + 4 \, e^{4} x^{3} - 16 \, 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*} \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, -2\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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