Optimal. Leaf size=66 \[ \frac{2 \sqrt{3} (2-e x)^{3/2}}{e}+\frac{24 \sqrt{3} \sqrt{2-e x}}{e}-\frac{48 \sqrt{3} \tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{e} \]
[Out]
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Rubi [A] time = 0.117541, antiderivative size = 66, 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 \[ \frac{2 \sqrt{3} (2-e x)^{3/2}}{e}+\frac{24 \sqrt{3} \sqrt{2-e x}}{e}-\frac{48 \sqrt{3} \tanh ^{-1}\left (\frac{1}{2} \sqrt{2-e x}\right )}{e} \]
Antiderivative was successfully verified.
[In] Int[(12 - 3*e^2*x^2)^(3/2)/(2 + e*x)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 13.7975, size = 54, normalized size = 0.82 \[ \frac{2 \left (- 3 e x + 6\right )^{\frac{3}{2}}}{3 e} + \frac{24 \sqrt{- 3 e x + 6}}{e} - \frac{48 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \sqrt{- 3 e x + 6}}{6} \right )}}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-3*e**2*x**2+12)**(3/2)/(e*x+2)**(5/2),x)
[Out]
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Mathematica [A] time = 0.0678738, size = 69, normalized size = 1.05 \[ -\frac{2 \sqrt{12-3 e^2 x^2} \left (\sqrt{e x-2} (e x-14)+24 \tan ^{-1}\left (\frac{1}{2} \sqrt{e x-2}\right )\right )}{e \sqrt{e x-2} \sqrt{e x+2}} \]
Antiderivative was successfully verified.
[In] Integrate[(12 - 3*e^2*x^2)^(3/2)/(2 + e*x)^(5/2),x]
[Out]
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Maple [A] time = 0.018, size = 77, normalized size = 1.2 \[ -2\,{\frac{\sqrt{-{e}^{2}{x}^{2}+4} \left ( xe\sqrt{-3\,ex+6}+24\,\sqrt{3}{\it Artanh} \left ( 1/6\,\sqrt{3}\sqrt{-3\,ex+6} \right ) -14\,\sqrt{-3\,ex+6} \right ) \sqrt{3}}{\sqrt{ex+2}\sqrt{-3\,ex+6}e}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-3*e^2*x^2+12)^(3/2)/(e*x+2)^(5/2),x)
[Out]
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Maxima [A] time = 0.878036, size = 61, normalized size = 0.92 \[ -\frac{2 \,{\left (i \, \sqrt{3}{\left (e x - 2\right )}^{\frac{3}{2}} + 24 i \, \sqrt{3} \arctan \left (\frac{1}{2} \, \sqrt{e x - 2}\right ) - 12 i \, \sqrt{3} \sqrt{e x - 2}\right )}}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-3*e^2*x^2 + 12)^(3/2)/(e*x + 2)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.217694, size = 169, normalized size = 2.56 \[ \frac{6 \,{\left (e^{3} x^{3} - 14 \, e^{2} x^{2} + 4 \, \sqrt{3} \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2} \log \left (-\frac{3 \, e^{2} x^{2} - 12 \, e x + 4 \, \sqrt{3} \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2} - 36}{e^{2} x^{2} + 4 \, e x + 4}\right ) - 4 \, e x + 56\right )}}{\sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2} e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-3*e^2*x^2 + 12)^(3/2)/(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((-3*e**2*x**2+12)**(3/2)/(e*x+2)**(5/2),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: AttributeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-3*e^2*x^2 + 12)^(3/2)/(e*x + 2)^(5/2),x, algorithm="giac")
[Out]