3.141 \(\int x^2 \sqrt{9-x^2} \, dx\)

Optimal. Leaf size=45 \[ -\frac{9}{8} \sqrt{9-x^2} x+\frac{1}{4} \sqrt{9-x^2} x^3+\frac{81}{8} \sin ^{-1}\left (\frac{x}{3}\right ) \]

[Out]

(-9*x*Sqrt[9 - x^2])/8 + (x^3*Sqrt[9 - x^2])/4 + (81*ArcSin[x/3])/8

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Rubi [A]  time = 0.0323906, antiderivative size = 45, 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 \[ -\frac{9}{8} \sqrt{9-x^2} x+\frac{1}{4} \sqrt{9-x^2} x^3+\frac{81}{8} \sin ^{-1}\left (\frac{x}{3}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^2*Sqrt[9 - x^2],x]

[Out]

(-9*x*Sqrt[9 - x^2])/8 + (x^3*Sqrt[9 - x^2])/4 + (81*ArcSin[x/3])/8

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Rubi in Sympy [A]  time = 2.77129, size = 34, normalized size = 0.76 \[ \frac{x^{3} \sqrt{- x^{2} + 9}}{4} - \frac{9 x \sqrt{- x^{2} + 9}}{8} + \frac{81 \operatorname{asin}{\left (\frac{x}{3} \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(-x**2+9)**(1/2),x)

[Out]

x**3*sqrt(-x**2 + 9)/4 - 9*x*sqrt(-x**2 + 9)/8 + 81*asin(x/3)/8

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Mathematica [A]  time = 0.0241363, size = 33, normalized size = 0.73 \[ \frac{1}{8} \left (x \sqrt{9-x^2} \left (2 x^2-9\right )+81 \sin ^{-1}\left (\frac{x}{3}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*Sqrt[9 - x^2],x]

[Out]

(x*Sqrt[9 - x^2]*(-9 + 2*x^2) + 81*ArcSin[x/3])/8

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Maple [A]  time = 0.007, size = 32, normalized size = 0.7 \[ -{\frac{x}{4} \left ( -{x}^{2}+9 \right ) ^{{\frac{3}{2}}}}+{\frac{9\,x}{8}\sqrt{-{x}^{2}+9}}+{\frac{81}{8}\arcsin \left ({\frac{x}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(-x^2+9)^(1/2),x)

[Out]

-1/4*x*(-x^2+9)^(3/2)+9/8*x*(-x^2+9)^(1/2)+81/8*arcsin(1/3*x)

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Maxima [A]  time = 1.50664, size = 42, normalized size = 0.93 \[ -\frac{1}{4} \,{\left (-x^{2} + 9\right )}^{\frac{3}{2}} x + \frac{9}{8} \, \sqrt{-x^{2} + 9} x + \frac{81}{8} \, \arcsin \left (\frac{1}{3} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^2 + 9)*x^2,x, algorithm="maxima")

[Out]

-1/4*(-x^2 + 9)^(3/2)*x + 9/8*sqrt(-x^2 + 9)*x + 81/8*arcsin(1/3*x)

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Fricas [A]  time = 0.209553, size = 166, normalized size = 3.69 \[ -\frac{24 \, x^{7} - 756 \, x^{5} + 6804 \, x^{3} + 162 \,{\left (x^{4} - 72 \, x^{2} + 12 \,{\left (x^{2} - 18\right )} \sqrt{-x^{2} + 9} + 648\right )} \arctan \left (\frac{\sqrt{-x^{2} + 9} - 3}{x}\right ) -{\left (2 \, x^{7} - 153 \, x^{5} + 1944 \, x^{3} - 5832 \, x\right )} \sqrt{-x^{2} + 9} - 17496 \, x}{8 \,{\left (x^{4} - 72 \, x^{2} + 12 \,{\left (x^{2} - 18\right )} \sqrt{-x^{2} + 9} + 648\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^2 + 9)*x^2,x, algorithm="fricas")

[Out]

-1/8*(24*x^7 - 756*x^5 + 6804*x^3 + 162*(x^4 - 72*x^2 + 12*(x^2 - 18)*sqrt(-x^2
+ 9) + 648)*arctan((sqrt(-x^2 + 9) - 3)/x) - (2*x^7 - 153*x^5 + 1944*x^3 - 5832*
x)*sqrt(-x^2 + 9) - 17496*x)/(x^4 - 72*x^2 + 12*(x^2 - 18)*sqrt(-x^2 + 9) + 648)

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Sympy [A]  time = 4.45745, size = 112, normalized size = 2.49 \[ \begin{cases} \frac{i x^{5}}{4 \sqrt{x^{2} - 9}} - \frac{27 i x^{3}}{8 \sqrt{x^{2} - 9}} + \frac{81 i x}{8 \sqrt{x^{2} - 9}} - \frac{81 i \operatorname{acosh}{\left (\frac{x}{3} \right )}}{8} & \text{for}\: \frac{\left |{x^{2}}\right |}{9} > 1 \\- \frac{x^{5}}{4 \sqrt{- x^{2} + 9}} + \frac{27 x^{3}}{8 \sqrt{- x^{2} + 9}} - \frac{81 x}{8 \sqrt{- x^{2} + 9}} + \frac{81 \operatorname{asin}{\left (\frac{x}{3} \right )}}{8} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(-x**2+9)**(1/2),x)

[Out]

Piecewise((I*x**5/(4*sqrt(x**2 - 9)) - 27*I*x**3/(8*sqrt(x**2 - 9)) + 81*I*x/(8*
sqrt(x**2 - 9)) - 81*I*acosh(x/3)/8, Abs(x**2)/9 > 1), (-x**5/(4*sqrt(-x**2 + 9)
) + 27*x**3/(8*sqrt(-x**2 + 9)) - 81*x/(8*sqrt(-x**2 + 9)) + 81*asin(x/3)/8, Tru
e))

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GIAC/XCAS [A]  time = 0.21574, size = 35, normalized size = 0.78 \[ \frac{1}{8} \,{\left (2 \, x^{2} - 9\right )} \sqrt{-x^{2} + 9} x + \frac{81}{8} \, \arcsin \left (\frac{1}{3} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^2 + 9)*x^2,x, algorithm="giac")

[Out]

1/8*(2*x^2 - 9)*sqrt(-x^2 + 9)*x + 81/8*arcsin(1/3*x)