Optimal. Leaf size=45 \[ -\frac{9}{32} \sqrt{9-4 x^2} x+\frac{1}{4} \sqrt{9-4 x^2} x^3+\frac{81}{64} \sin ^{-1}\left (\frac{2 x}{3}\right ) \]
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Rubi [A] time = 0.0396414, 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 \[ -\frac{9}{32} \sqrt{9-4 x^2} x+\frac{1}{4} \sqrt{9-4 x^2} x^3+\frac{81}{64} \sin ^{-1}\left (\frac{2 x}{3}\right ) \]
Antiderivative was successfully verified.
[In] Int[x^2*Sqrt[9 - 4*x^2],x]
[Out]
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Rubi in Sympy [A] time = 5.93338, size = 39, normalized size = 0.87 \[ \frac{x^{3} \sqrt{- 4 x^{2} + 9}}{4} - \frac{9 x \sqrt{- 4 x^{2} + 9}}{32} + \frac{81 \operatorname{asin}{\left (\frac{2 x}{3} \right )}}{64} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(-4*x**2+9)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0276222, size = 36, normalized size = 0.8 \[ \sqrt{9-4 x^2} \left (\frac{x^3}{4}-\frac{9 x}{32}\right )+\frac{81}{64} \sin ^{-1}\left (\frac{2 x}{3}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x^2*Sqrt[9 - 4*x^2],x]
[Out]
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Maple [A] time = 0.008, size = 32, normalized size = 0.7 \[ -{\frac{x}{16} \left ( -4\,{x}^{2}+9 \right ) ^{{\frac{3}{2}}}}+{\frac{9\,x}{32}\sqrt{-4\,{x}^{2}+9}}+{\frac{81}{64}\arcsin \left ({\frac{2\,x}{3}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(-4*x^2+9)^(1/2),x)
[Out]
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Maxima [A] time = 1.50466, size = 42, normalized size = 0.93 \[ -\frac{1}{16} \,{\left (-4 \, x^{2} + 9\right )}^{\frac{3}{2}} x + \frac{9}{32} \, \sqrt{-4 \, x^{2} + 9} x + \frac{81}{64} \, \arcsin \left (\frac{2}{3} \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-4*x^2 + 9)*x^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.231001, size = 178, normalized size = 3.96 \[ -\frac{192 \, x^{7} - 1512 \, x^{5} + 3402 \, x^{3} + 81 \,{\left (2 \, x^{4} - 36 \, x^{2} + 3 \,{\left (2 \, x^{2} - 9\right )} \sqrt{-4 \, x^{2} + 9} + 81\right )} \arctan \left (\frac{\sqrt{-4 \, x^{2} + 9} - 3}{2 \, x}\right ) -{\left (16 \, x^{7} - 306 \, x^{5} + 972 \, x^{3} - 729 \, x\right )} \sqrt{-4 \, x^{2} + 9} - 2187 \, x}{32 \,{\left (2 \, x^{4} - 36 \, x^{2} + 3 \,{\left (2 \, x^{2} - 9\right )} \sqrt{-4 \, x^{2} + 9} + 81\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-4*x^2 + 9)*x^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 9.01955, size = 124, normalized size = 2.76 \[ \begin{cases} \frac{i x^{5}}{\sqrt{4 x^{2} - 9}} - \frac{27 i x^{3}}{8 \sqrt{4 x^{2} - 9}} + \frac{81 i x}{32 \sqrt{4 x^{2} - 9}} - \frac{81 i \operatorname{acosh}{\left (\frac{2 x}{3} \right )}}{64} & \text{for}\: \frac{4 \left |{x^{2}}\right |}{9} > 1 \\- \frac{x^{5}}{\sqrt{- 4 x^{2} + 9}} + \frac{27 x^{3}}{8 \sqrt{- 4 x^{2} + 9}} - \frac{81 x}{32 \sqrt{- 4 x^{2} + 9}} + \frac{81 \operatorname{asin}{\left (\frac{2 x}{3} \right )}}{64} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2*(-4*x**2+9)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.208278, size = 35, normalized size = 0.78 \[ \frac{1}{32} \,{\left (8 \, x^{2} - 9\right )} \sqrt{-4 \, x^{2} + 9} x + \frac{81}{64} \, \arcsin \left (\frac{2}{3} \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-4*x^2 + 9)*x^2,x, algorithm="giac")
[Out]