Optimal. Leaf size=99 \[ -\frac{7}{135} \sqrt{2-3 x^2} \sqrt{3 x^2-1} x-\frac{1}{15} \sqrt{2-3 x^2} \sqrt{3 x^2-1} x^3-\frac{2 F\left (\left .\cos ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{27 \sqrt{3}}-\frac{8 E\left (\left .\cos ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{45 \sqrt{3}} \]
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Rubi [A] time = 0.298323, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{7}{135} \sqrt{2-3 x^2} \sqrt{3 x^2-1} x-\frac{1}{15} \sqrt{2-3 x^2} \sqrt{3 x^2-1} x^3-\frac{2 F\left (\left .\cos ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{27 \sqrt{3}}-\frac{8 E\left (\left .\cos ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{45 \sqrt{3}} \]
Antiderivative was successfully verified.
[In] Int[(x^4*Sqrt[-1 + 3*x^2])/Sqrt[2 - 3*x^2],x]
[Out]
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Rubi in Sympy [A] time = 34.8414, size = 92, normalized size = 0.93 \[ - \frac{x^{3} \sqrt{- 3 x^{2} + 2} \sqrt{3 x^{2} - 1}}{15} - \frac{7 x \sqrt{- 3 x^{2} + 2} \sqrt{3 x^{2} - 1}}{135} - \frac{8 \sqrt{3} E\left (\operatorname{acos}{\left (\frac{\sqrt{6} x}{2} \right )}\middle | 2\right )}{135} - \frac{2 \sqrt{3} F\left (\operatorname{acos}{\left (\frac{\sqrt{6} x}{2} \right )}\middle | 2\right )}{81} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4*(3*x**2-1)**(1/2)/(-3*x**2+2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.143801, size = 92, normalized size = 0.93 \[ \frac{10 \sqrt{3-9 x^2} F\left (\left .\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )-24 \sqrt{3-9 x^2} E\left (\left .\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )-3 x \sqrt{2-3 x^2} \left (27 x^4+12 x^2-7\right )}{405 \sqrt{3 x^2-1}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^4*Sqrt[-1 + 3*x^2])/Sqrt[2 - 3*x^2],x]
[Out]
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Maple [A] time = 0.031, size = 135, normalized size = 1.4 \[ -{\frac{\sqrt{2}}{7290\,{x}^{4}-7290\,{x}^{2}+1620}\sqrt{3\,{x}^{2}-1}\sqrt{-6\,{x}^{2}+4} \left ( 243\,{x}^{7}-54\,{x}^{5}+5\,\sqrt{3}\sqrt{2}\sqrt{-6\,{x}^{2}+4}\sqrt{-3\,{x}^{2}+1}{\it EllipticF} \left ( 1/2\,x\sqrt{3}\sqrt{2},\sqrt{2} \right ) -12\,\sqrt{3}\sqrt{2}\sqrt{-6\,{x}^{2}+4}\sqrt{-3\,{x}^{2}+1}{\it EllipticE} \left ( 1/2\,x\sqrt{3}\sqrt{2},\sqrt{2} \right ) -135\,{x}^{3}+42\,x \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4*(3*x^2-1)^(1/2)/(-3*x^2+2)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{3 \, x^{2} - 1} x^{4}}{\sqrt{-3 \, x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(3*x^2 - 1)*x^4/sqrt(-3*x^2 + 2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{3 \, x^{2} - 1} x^{4}}{\sqrt{-3 \, x^{2} + 2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(3*x^2 - 1)*x^4/sqrt(-3*x^2 + 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(x**4*(3*x**2-1)**(1/2)/(-3*x**2+2)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{3 \, x^{2} - 1} x^{4}}{\sqrt{-3 \, x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(3*x^2 - 1)*x^4/sqrt(-3*x^2 + 2),x, algorithm="giac")
[Out]