Optimal. Leaf size=141 \[ -\frac{2 \left (4-e^2 x^2\right )^{5/4}}{1105\ 3^{3/4} e (e x+2)^{5/2}}-\frac{2 \left (4-e^2 x^2\right )^{5/4}}{221\ 3^{3/4} e (e x+2)^{7/2}}-\frac{3 \sqrt [4]{3} \left (4-e^2 x^2\right )^{5/4}}{221 e (e x+2)^{9/2}}-\frac{\sqrt [4]{3} \left (4-e^2 x^2\right )^{5/4}}{17 e (e x+2)^{11/2}} \]
[Out]
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Rubi [A] time = 0.194923, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{2 \left (4-e^2 x^2\right )^{5/4}}{1105\ 3^{3/4} e (e x+2)^{5/2}}-\frac{2 \left (4-e^2 x^2\right )^{5/4}}{221\ 3^{3/4} e (e x+2)^{7/2}}-\frac{3 \sqrt [4]{3} \left (4-e^2 x^2\right )^{5/4}}{221 e (e x+2)^{9/2}}-\frac{\sqrt [4]{3} \left (4-e^2 x^2\right )^{5/4}}{17 e (e x+2)^{11/2}} \]
Antiderivative was successfully verified.
[In] Int[(12 - 3*e^2*x^2)^(1/4)/(2 + e*x)^(11/2),x]
[Out]
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Rubi in Sympy [A] time = 13.3661, size = 105, normalized size = 0.74 \[ - \frac{2 \left (- 3 e^{2} x^{2} + 12\right )^{\frac{5}{4}}}{9945 e \left (e x + 2\right )^{\frac{5}{2}}} - \frac{2 \left (- 3 e^{2} x^{2} + 12\right )^{\frac{5}{4}}}{1989 e \left (e x + 2\right )^{\frac{7}{2}}} - \frac{\left (- 3 e^{2} x^{2} + 12\right )^{\frac{5}{4}}}{221 e \left (e x + 2\right )^{\frac{9}{2}}} - \frac{\left (- 3 e^{2} x^{2} + 12\right )^{\frac{5}{4}}}{51 e \left (e x + 2\right )^{\frac{11}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-3*e**2*x**2+12)**(1/4)/(e*x+2)**(11/2),x)
[Out]
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Mathematica [A] time = 0.0567826, size = 65, normalized size = 0.46 \[ \frac{\sqrt [4]{4-e^2 x^2} \left (2 e^4 x^4+18 e^3 x^3+65 e^2 x^2+123 e x-682\right )}{1105\ 3^{3/4} e (e x+2)^{9/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(12 - 3*e^2*x^2)^(1/4)/(2 + e*x)^(11/2),x]
[Out]
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Maple [A] time = 0.008, size = 52, normalized size = 0.4 \[{\frac{ \left ( ex-2 \right ) \left ( 2\,{e}^{3}{x}^{3}+22\,{e}^{2}{x}^{2}+109\,ex+341 \right ) }{3315\,e}\sqrt [4]{-3\,{e}^{2}{x}^{2}+12} \left ( ex+2 \right ) ^{-{\frac{9}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-3*e^2*x^2+12)^(1/4)/(e*x+2)^(11/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}}}{{\left (e x + 2\right )}^{\frac{11}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-3*e^2*x^2 + 12)^(1/4)/(e*x + 2)^(11/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.222472, size = 127, normalized size = 0.9 \[ \frac{{\left (2 \, e^{4} x^{4} + 18 \, e^{3} x^{3} + 65 \, e^{2} x^{2} + 123 \, e x - 682\right )}{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}} \sqrt{e x + 2}}{3315 \,{\left (e^{6} x^{5} + 10 \, e^{5} x^{4} + 40 \, e^{4} x^{3} + 80 \, e^{3} x^{2} + 80 \, e^{2} x + 32 \, e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-3*e^2*x^2 + 12)^(1/4)/(e*x + 2)^(11/2),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)**(1/4)/(e*x+2)**(11/2),x)
[Out]
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GIAC/XCAS [A] time = 0.249938, size = 284, normalized size = 2.01 \[ -\frac{1}{212160} \cdot 3^{\frac{1}{4}}{\left (\frac{663 \,{\left (-{\left (x e + 2\right )}^{2} + 4 \, x e + 8\right )}^{\frac{1}{4}}{\left (\frac{4}{x e + 2} - 1\right )}}{\sqrt{x e + 2}} + \frac{1105 \,{\left ({\left (x e + 2\right )}^{2} - 8 \, x e\right )}{\left (-{\left (x e + 2\right )}^{2} + 4 \, x e + 8\right )}^{\frac{1}{4}}}{{\left (x e + 2\right )}^{\frac{5}{2}}} - \frac{765 \,{\left ({\left (x e + 2\right )}^{3} - 12 \,{\left (x e + 2\right )}^{2} + 48 \, x e + 32\right )}{\left (-{\left (x e + 2\right )}^{2} + 4 \, x e + 8\right )}^{\frac{1}{4}}}{{\left (x e + 2\right )}^{\frac{7}{2}}} + \frac{195 \,{\left ({\left (x e + 2\right )}^{4} - 16 \,{\left (x e + 2\right )}^{3} + 96 \,{\left (x e + 2\right )}^{2} - 256 \, x e - 256\right )}{\left (-{\left (x e + 2\right )}^{2} + 4 \, x e + 8\right )}^{\frac{1}{4}}}{{\left (x e + 2\right )}^{\frac{9}{2}}}\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-3*e^2*x^2 + 12)^(1/4)/(e*x + 2)^(11/2),x, algorithm="giac")
[Out]