3.455 \(\int x^2 \sqrt{9-4 x^2} \, dx\)

Optimal. Leaf size=45 \[ -\frac{9}{32} \sqrt{9-4 x^2} x+\frac{1}{4} \sqrt{9-4 x^2} x^3+\frac{81}{64} \sin ^{-1}\left (\frac{2 x}{3}\right ) \]

[Out]

(-9*x*Sqrt[9 - 4*x^2])/32 + (x^3*Sqrt[9 - 4*x^2])/4 + (81*ArcSin[(2*x)/3])/64

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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]

(-9*x*Sqrt[9 - 4*x^2])/32 + (x^3*Sqrt[9 - 4*x^2])/4 + (81*ArcSin[(2*x)/3])/64

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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]

x**3*sqrt(-4*x**2 + 9)/4 - 9*x*sqrt(-4*x**2 + 9)/32 + 81*asin(2*x/3)/64

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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]

Sqrt[9 - 4*x^2]*((-9*x)/32 + x^3/4) + (81*ArcSin[(2*x)/3])/64

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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]

-1/16*x*(-4*x^2+9)^(3/2)+9/32*x*(-4*x^2+9)^(1/2)+81/64*arcsin(2/3*x)

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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]

-1/16*(-4*x^2 + 9)^(3/2)*x + 9/32*sqrt(-4*x^2 + 9)*x + 81/64*arcsin(2/3*x)

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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]

-1/32*(192*x^7 - 1512*x^5 + 3402*x^3 + 81*(2*x^4 - 36*x^2 + 3*(2*x^2 - 9)*sqrt(-
4*x^2 + 9) + 81)*arctan(1/2*(sqrt(-4*x^2 + 9) - 3)/x) - (16*x^7 - 306*x^5 + 972*
x^3 - 729*x)*sqrt(-4*x^2 + 9) - 2187*x)/(2*x^4 - 36*x^2 + 3*(2*x^2 - 9)*sqrt(-4*
x^2 + 9) + 81)

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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]

Piecewise((I*x**5/sqrt(4*x**2 - 9) - 27*I*x**3/(8*sqrt(4*x**2 - 9)) + 81*I*x/(32
*sqrt(4*x**2 - 9)) - 81*I*acosh(2*x/3)/64, 4*Abs(x**2)/9 > 1), (-x**5/sqrt(-4*x*
*2 + 9) + 27*x**3/(8*sqrt(-4*x**2 + 9)) - 81*x/(32*sqrt(-4*x**2 + 9)) + 81*asin(
2*x/3)/64, True))

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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]

1/32*(8*x^2 - 9)*sqrt(-4*x^2 + 9)*x + 81/64*arcsin(2/3*x)