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