3.369 \(\int x^2 \sqrt{5-x^2} \, dx\)

Optimal. Leaf size=47 \[ -\frac{5}{8} \sqrt{5-x^2} x+\frac{1}{4} \sqrt{5-x^2} x^3+\frac{25}{8} \sin ^{-1}\left (\frac{x}{\sqrt{5}}\right ) \]

[Out]

(-5*x*Sqrt[5 - x^2])/8 + (x^3*Sqrt[5 - x^2])/4 + (25*ArcSin[x/Sqrt[5]])/8

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Rubi [A]  time = 0.0337825, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{5}{8} \sqrt{5-x^2} x+\frac{1}{4} \sqrt{5-x^2} x^3+\frac{25}{8} \sin ^{-1}\left (\frac{x}{\sqrt{5}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^2*Sqrt[5 - x^2],x]

[Out]

(-5*x*Sqrt[5 - x^2])/8 + (x^3*Sqrt[5 - x^2])/4 + (25*ArcSin[x/Sqrt[5]])/8

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Rubi in Sympy [A]  time = 2.79844, size = 39, normalized size = 0.83 \[ \frac{x^{3} \sqrt{- x^{2} + 5}}{4} - \frac{5 x \sqrt{- x^{2} + 5}}{8} + \frac{25 \operatorname{asin}{\left (\frac{\sqrt{5} x}{5} \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(-x**2+5)**(1/2),x)

[Out]

x**3*sqrt(-x**2 + 5)/4 - 5*x*sqrt(-x**2 + 5)/8 + 25*asin(sqrt(5)*x/5)/8

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Mathematica [A]  time = 0.0303433, size = 35, normalized size = 0.74 \[ \frac{1}{8} \left (x \sqrt{5-x^2} \left (2 x^2-5\right )+25 \sin ^{-1}\left (\frac{x}{\sqrt{5}}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*Sqrt[5 - x^2],x]

[Out]

(x*Sqrt[5 - x^2]*(-5 + 2*x^2) + 25*ArcSin[x/Sqrt[5]])/8

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Maple [A]  time = 0.01, size = 35, normalized size = 0.7 \[ -{\frac{x}{4} \left ( -{x}^{2}+5 \right ) ^{{\frac{3}{2}}}}+{\frac{5\,x}{8}\sqrt{-{x}^{2}+5}}+{\frac{25}{8}\arcsin \left ({\frac{x\sqrt{5}}{5}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(-x^2+5)^(1/2),x)

[Out]

-1/4*x*(-x^2+5)^(3/2)+5/8*x*(-x^2+5)^(1/2)+25/8*arcsin(1/5*x*5^(1/2))

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Maxima [A]  time = 1.51772, size = 46, normalized size = 0.98 \[ -\frac{1}{4} \,{\left (-x^{2} + 5\right )}^{\frac{3}{2}} x + \frac{5}{8} \, \sqrt{-x^{2} + 5} x + \frac{25}{8} \, \arcsin \left (\frac{1}{5} \, \sqrt{5} x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^2 + 5)*x^2,x, algorithm="maxima")

[Out]

-1/4*(-x^2 + 5)^(3/2)*x + 5/8*sqrt(-x^2 + 5)*x + 25/8*arcsin(1/5*sqrt(5)*x)

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Fricas [A]  time = 0.233044, size = 50, normalized size = 1.06 \[ \frac{1}{8} \,{\left (2 \, x^{3} - 5 \, x\right )} \sqrt{-x^{2} + 5} - \frac{25}{8} \, \arctan \left (\frac{\sqrt{-x^{2} + 5}}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^2 + 5)*x^2,x, algorithm="fricas")

[Out]

1/8*(2*x^3 - 5*x)*sqrt(-x^2 + 5) - 25/8*arctan(sqrt(-x^2 + 5)/x)

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Sympy [A]  time = 4.47294, size = 122, normalized size = 2.6 \[ \begin{cases} \frac{i x^{5}}{4 \sqrt{x^{2} - 5}} - \frac{15 i x^{3}}{8 \sqrt{x^{2} - 5}} + \frac{25 i x}{8 \sqrt{x^{2} - 5}} - \frac{25 i \operatorname{acosh}{\left (\frac{\sqrt{5} x}{5} \right )}}{8} & \text{for}\: \frac{\left |{x^{2}}\right |}{5} > 1 \\- \frac{x^{5}}{4 \sqrt{- x^{2} + 5}} + \frac{15 x^{3}}{8 \sqrt{- x^{2} + 5}} - \frac{25 x}{8 \sqrt{- x^{2} + 5}} + \frac{25 \operatorname{asin}{\left (\frac{\sqrt{5} x}{5} \right )}}{8} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(-x**2+5)**(1/2),x)

[Out]

Piecewise((I*x**5/(4*sqrt(x**2 - 5)) - 15*I*x**3/(8*sqrt(x**2 - 5)) + 25*I*x/(8*
sqrt(x**2 - 5)) - 25*I*acosh(sqrt(5)*x/5)/8, Abs(x**2)/5 > 1), (-x**5/(4*sqrt(-x
**2 + 5)) + 15*x**3/(8*sqrt(-x**2 + 5)) - 25*x/(8*sqrt(-x**2 + 5)) + 25*asin(sqr
t(5)*x/5)/8, True))

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GIAC/XCAS [A]  time = 0.208678, size = 39, normalized size = 0.83 \[ \frac{1}{8} \,{\left (2 \, x^{2} - 5\right )} \sqrt{-x^{2} + 5} x + \frac{25}{8} \, \arcsin \left (\frac{1}{5} \, \sqrt{5} x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^2 + 5)*x^2,x, algorithm="giac")

[Out]

1/8*(2*x^2 - 5)*sqrt(-x^2 + 5)*x + 25/8*arcsin(1/5*sqrt(5)*x)