Optimal. Leaf size=86 \[ \frac{\sqrt{3} \sqrt{2-e x}}{16 e (e x+2)}-\frac{\sqrt{3} \sqrt{2-e x}}{2 e (e x+2)^2}+\frac{\sqrt{3} \tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{32 e} \]
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Rubi [A] time = 0.0290316, antiderivative size = 86, 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, 51, 63, 206} \[ \frac{\sqrt{3} \sqrt{2-e x}}{16 e (e x+2)}-\frac{\sqrt{3} \sqrt{2-e x}}{2 e (e x+2)^2}+\frac{\sqrt{3} \tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{32 e} \]
Antiderivative was successfully verified.
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Rule 627
Rule 47
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{\sqrt{12-3 e^2 x^2}}{(2+e x)^{7/2}} \, dx &=\int \frac{\sqrt{6-3 e x}}{(2+e x)^3} \, dx\\ &=-\frac{\sqrt{3} \sqrt{2-e x}}{2 e (2+e x)^2}-\frac{3}{4} \int \frac{1}{\sqrt{6-3 e x} (2+e x)^2} \, dx\\ &=-\frac{\sqrt{3} \sqrt{2-e x}}{2 e (2+e x)^2}+\frac{\sqrt{3} \sqrt{2-e x}}{16 e (2+e x)}-\frac{3}{32} \int \frac{1}{\sqrt{6-3 e x} (2+e x)} \, dx\\ &=-\frac{\sqrt{3} \sqrt{2-e x}}{2 e (2+e x)^2}+\frac{\sqrt{3} \sqrt{2-e x}}{16 e (2+e x)}+\frac{\operatorname{Subst}\left (\int \frac{1}{4-\frac{x^2}{3}} \, dx,x,\sqrt{6-3 e x}\right )}{16 e}\\ &=-\frac{\sqrt{3} \sqrt{2-e x}}{2 e (2+e x)^2}+\frac{\sqrt{3} \sqrt{2-e x}}{16 e (2+e x)}+\frac{\sqrt{3} \tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{32 e}\\ \end{align*}
Mathematica [C] time = 0.0679962, size = 54, normalized size = 0.63 \[ \frac{(e x-2) \sqrt{4-e^2 x^2} \, _2F_1\left (\frac{3}{2},3;\frac{5}{2};\frac{1}{2}-\frac{e x}{4}\right )}{32 e \sqrt{3 e x+6}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.15, size = 125, normalized size = 1.5 \begin{align*}{\frac{\sqrt{3}}{32\,e}\sqrt{-{e}^{2}{x}^{2}+4} \left ( \sqrt{3}{\it Artanh} \left ({\frac{\sqrt{3}}{6}\sqrt{-3\,ex+6}} \right ){x}^{2}{e}^{2}+4\,\sqrt{3}{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ) xe+2\,xe\sqrt{-3\,ex+6}+4\,\sqrt{3}{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ) -12\,\sqrt{-3\,ex+6} \right ){\frac{1}{\sqrt{ \left ( ex+2 \right ) ^{5}}}}{\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{\sqrt{-3 \, e^{2} x^{2} + 12}}{{\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 = 1.87538, size = 327, normalized size = 3.8 \begin{align*} \frac{\sqrt{3}{\left (e^{3} x^{3} + 6 \, e^{2} x^{2} + 12 \, e x + 8\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 \, \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2}{\left (e x - 6\right )}}{64 \,{\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{\sqrt{-3 \, e^{2} x^{2} + 12}}{{\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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