Optimal. Leaf size=65 \[ -\frac{25}{16} \sqrt{5-x^2} x+\frac{1}{6} \sqrt{5-x^2} x^5-\frac{5}{24} \sqrt{5-x^2} x^3+\frac{125}{16} \sin ^{-1}\left (\frac{x}{\sqrt{5}}\right ) \]
[Out]
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Rubi [A] time = 0.0544259, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{25}{16} \sqrt{5-x^2} x+\frac{1}{6} \sqrt{5-x^2} x^5-\frac{5}{24} \sqrt{5-x^2} x^3+\frac{125}{16} \sin ^{-1}\left (\frac{x}{\sqrt{5}}\right ) \]
Antiderivative was successfully verified.
[In] Int[x^4*Sqrt[5 - x^2],x]
[Out]
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Rubi in Sympy [A] time = 3.6922, size = 54, normalized size = 0.83 \[ \frac{x^{5} \sqrt{- x^{2} + 5}}{6} - \frac{5 x^{3} \sqrt{- x^{2} + 5}}{24} - \frac{25 x \sqrt{- x^{2} + 5}}{16} + \frac{125 \operatorname{asin}{\left (\frac{\sqrt{5} x}{5} \right )}}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4*(-x**2+5)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0368336, size = 40, normalized size = 0.62 \[ \frac{1}{48} \left (x \sqrt{5-x^2} \left (8 x^4-10 x^2-75\right )+375 \sin ^{-1}\left (\frac{x}{\sqrt{5}}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x^4*Sqrt[5 - x^2],x]
[Out]
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Maple [A] time = 0.009, size = 49, normalized size = 0.8 \[ -{\frac{{x}^{3}}{6} \left ( -{x}^{2}+5 \right ) ^{{\frac{3}{2}}}}-{\frac{5\,x}{8} \left ( -{x}^{2}+5 \right ) ^{{\frac{3}{2}}}}+{\frac{25\,x}{16}\sqrt{-{x}^{2}+5}}+{\frac{125}{16}\arcsin \left ({\frac{x\sqrt{5}}{5}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4*(-x^2+5)^(1/2),x)
[Out]
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Maxima [A] time = 1.49936, size = 65, normalized size = 1. \[ -\frac{1}{6} \,{\left (-x^{2} + 5\right )}^{\frac{3}{2}} x^{3} - \frac{5}{8} \,{\left (-x^{2} + 5\right )}^{\frac{3}{2}} x + \frac{25}{16} \, \sqrt{-x^{2} + 5} x + \frac{125}{16} \, \arcsin \left (\frac{1}{5} \, \sqrt{5} x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x^2 + 5)*x^4,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.219146, size = 57, normalized size = 0.88 \[ \frac{1}{48} \,{\left (8 \, x^{5} - 10 \, x^{3} - 75 \, x\right )} \sqrt{-x^{2} + 5} - \frac{125}{16} \, \arctan \left (\frac{\sqrt{-x^{2} + 5}}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x^2 + 5)*x^4,x, algorithm="fricas")
[Out]
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Sympy [A] time = 7.68795, size = 155, normalized size = 2.38 \[ \begin{cases} \frac{i x^{7}}{6 \sqrt{x^{2} - 5}} - \frac{25 i x^{5}}{24 \sqrt{x^{2} - 5}} - \frac{25 i x^{3}}{48 \sqrt{x^{2} - 5}} + \frac{125 i x}{16 \sqrt{x^{2} - 5}} - \frac{125 i \operatorname{acosh}{\left (\frac{\sqrt{5} x}{5} \right )}}{16} & \text{for}\: \frac{\left |{x^{2}}\right |}{5} > 1 \\- \frac{x^{7}}{6 \sqrt{- x^{2} + 5}} + \frac{25 x^{5}}{24 \sqrt{- x^{2} + 5}} + \frac{25 x^{3}}{48 \sqrt{- x^{2} + 5}} - \frac{125 x}{16 \sqrt{- x^{2} + 5}} + \frac{125 \operatorname{asin}{\left (\frac{\sqrt{5} x}{5} \right )}}{16} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4*(-x**2+5)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.210067, size = 49, normalized size = 0.75 \[ \frac{1}{48} \,{\left (2 \,{\left (4 \, x^{2} - 5\right )} x^{2} - 75\right )} \sqrt{-x^{2} + 5} x + \frac{125}{16} \, \arcsin \left (\frac{1}{5} \, \sqrt{5} x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x^2 + 5)*x^4,x, algorithm="giac")
[Out]