Optimal. Leaf size=86 \[ -\frac{\sqrt{3} \sqrt{2-e x}}{64 e (e x+2)}-\frac{\sqrt{2-e x}}{8 \sqrt{3} e (e x+2)^2}-\frac{\sqrt{3} \tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{128 e} \]
[Out]
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Rubi [A] time = 0.125842, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{\sqrt{3} \sqrt{2-e x}}{64 e (e x+2)}-\frac{\sqrt{2-e x}}{8 \sqrt{3} e (e x+2)^2}-\frac{\sqrt{3} \tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{128 e} \]
Antiderivative was successfully verified.
[In] Int[1/((2 + e*x)^(5/2)*Sqrt[12 - 3*e^2*x^2]),x]
[Out]
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Rubi in Sympy [A] time = 14.3813, size = 66, normalized size = 0.77 \[ - \frac{\sqrt{- 3 e x + 6}}{64 e \left (e x + 2\right )} - \frac{\sqrt{- 3 e x + 6}}{24 e \left (e x + 2\right )^{2}} - \frac{\sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \sqrt{- 3 e x + 6}}{6} \right )}}{128 e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(e*x+2)**(5/2)/(-3*e**2*x**2+12)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0941284, size = 76, normalized size = 0.88 \[ \frac{6 e^2 x^2+16 e x+3 \sqrt{e x-2} (e x+2)^2 \tan ^{-1}\left (\frac{1}{2} \sqrt{e x-2}\right )-56}{128 e (e x+2)^{3/2} \sqrt{12-3 e^2 x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((2 + e*x)^(5/2)*Sqrt[12 - 3*e^2*x^2]),x]
[Out]
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Maple [A] time = 0.021, size = 126, normalized size = 1.5 \[ -{\frac{\sqrt{3}}{384\,e}\sqrt{-{e}^{2}{x}^{2}+4} \left ( 3\,{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ) \sqrt{3}{x}^{2}{e}^{2}+12\,{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ) \sqrt{3}xe+6\,xe\sqrt{-3\,ex+6}+12\,\sqrt{3}{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ) +28\,\sqrt{-3\,ex+6} \right ){\frac{1}{\sqrt{ \left ( ex+2 \right ) ^{5}}}}{\frac{1}{\sqrt{-3\,ex+6}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(e*x+2)^(5/2)/(-3*e^2*x^2+12)^(1/2),x)
[Out]
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Maxima [A] time = 0.839566, size = 97, normalized size = 1.13 \[ -\frac{i \, \sqrt{3} \arctan \left (\frac{1}{2} \, \sqrt{e x - 2}\right ) - \frac{256 \,{\left (3 \,{\left (e x - 2\right )}^{\frac{3}{2}} + 20 \, \sqrt{e x - 2}\right )}}{128 i \, \sqrt{3}{\left (e x - 2\right )}^{2} + 1024 i \, \sqrt{3}{\left (e x - 2\right )} + 2048 i \, \sqrt{3}}}{128 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(-3*e^2*x^2 + 12)*(e*x + 2)^(5/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.224518, size = 188, normalized size = 2.19 \[ \frac{3 \, \sqrt{3}{\left (e^{3} x^{3} + 6 \, e^{2} x^{2} + 12 \, e x + 8\right )} \log \left (-\frac{3 \, e^{2} x^{2} - 12 \, e x + 4 \, \sqrt{3} \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2} - 36}{e^{2} x^{2} + 4 \, e x + 4}\right ) - 4 \, \sqrt{-3 \, e^{2} x^{2} + 12}{\left (3 \, e x + 14\right )} \sqrt{e x + 2}}{768 \,{\left (e^{4} x^{3} + 6 \, e^{3} x^{2} + 12 \, e^{2} x + 8 \, e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(-3*e^2*x^2 + 12)*(e*x + 2)^(5/2)),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(e*x+2)**(5/2)/(-3*e**2*x**2+12)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.617204, size = 4, normalized size = 0.05 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(-3*e^2*x^2 + 12)*(e*x + 2)^(5/2)),x, algorithm="giac")
[Out]