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

Optimal. Leaf size=45 \[ -\frac{27}{128} \sqrt{9-4 x^2} x-\frac{1}{16} \sqrt{9-4 x^2} x^3+\frac{243}{256} \sin ^{-1}\left (\frac{2 x}{3}\right ) \]

[Out]

(-27*x*Sqrt[9 - 4*x^2])/128 - (x^3*Sqrt[9 - 4*x^2])/16 + (243*ArcSin[(2*x)/3])/2
56

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Rubi [A]  time = 0.040779, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{27}{128} \sqrt{9-4 x^2} x-\frac{1}{16} \sqrt{9-4 x^2} x^3+\frac{243}{256} \sin ^{-1}\left (\frac{2 x}{3}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^4/Sqrt[9 - 4*x^2],x]

[Out]

(-27*x*Sqrt[9 - 4*x^2])/128 - (x^3*Sqrt[9 - 4*x^2])/16 + (243*ArcSin[(2*x)/3])/2
56

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Rubi in Sympy [A]  time = 5.48068, size = 39, normalized size = 0.87 \[ - \frac{x^{3} \sqrt{- 4 x^{2} + 9}}{16} - \frac{27 x \sqrt{- 4 x^{2} + 9}}{128} + \frac{243 \operatorname{asin}{\left (\frac{2 x}{3} \right )}}{256} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(-4*x**2+9)**(1/2),x)

[Out]

-x**3*sqrt(-4*x**2 + 9)/16 - 27*x*sqrt(-4*x**2 + 9)/128 + 243*asin(2*x/3)/256

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Mathematica [A]  time = 0.0308707, size = 36, normalized size = 0.8 \[ \sqrt{9-4 x^2} \left (-\frac{x^3}{16}-\frac{27 x}{128}\right )+\frac{243}{256} \sin ^{-1}\left (\frac{2 x}{3}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/Sqrt[9 - 4*x^2],x]

[Out]

Sqrt[9 - 4*x^2]*((-27*x)/128 - x^3/16) + (243*ArcSin[(2*x)/3])/256

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Maple [A]  time = 0.01, size = 34, normalized size = 0.8 \[{\frac{243}{256}\arcsin \left ({\frac{2\,x}{3}} \right ) }-{\frac{27\,x}{128}\sqrt{-4\,{x}^{2}+9}}-{\frac{{x}^{3}}{16}\sqrt{-4\,{x}^{2}+9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(-4*x^2+9)^(1/2),x)

[Out]

243/256*arcsin(2/3*x)-27/128*x*(-4*x^2+9)^(1/2)-1/16*x^3*(-4*x^2+9)^(1/2)

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Maxima [A]  time = 1.53899, size = 45, normalized size = 1. \[ -\frac{1}{16} \, \sqrt{-4 \, x^{2} + 9} x^{3} - \frac{27}{128} \, \sqrt{-4 \, x^{2} + 9} x + \frac{243}{256} \, \arcsin \left (\frac{2}{3} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/sqrt(-4*x^2 + 9),x, algorithm="maxima")

[Out]

-1/16*sqrt(-4*x^2 + 9)*x^3 - 27/128*sqrt(-4*x^2 + 9)*x + 243/256*arcsin(2/3*x)

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Fricas [A]  time = 0.230018, size = 178, normalized size = 3.96 \[ \frac{192 \, x^{7} - 648 \, x^{5} - 2430 \, x^{3} - 243 \,{\left (2 \, x^{4} - 36 \, x^{2} + 3 \,{\left (2 \, x^{2} - 9\right )} \sqrt{-4 \, x^{2} + 9} + 81\right )} \arctan \left (\frac{\sqrt{-4 \, x^{2} + 9} - 3}{2 \, x}\right ) -{\left (16 \, x^{7} - 234 \, x^{5} - 324 \, x^{3} + 2187 \, x\right )} \sqrt{-4 \, x^{2} + 9} + 6561 \, x}{128 \,{\left (2 \, x^{4} - 36 \, x^{2} + 3 \,{\left (2 \, x^{2} - 9\right )} \sqrt{-4 \, x^{2} + 9} + 81\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/sqrt(-4*x^2 + 9),x, algorithm="fricas")

[Out]

1/128*(192*x^7 - 648*x^5 - 2430*x^3 - 243*(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 - 234*x^5 - 324*
x^3 + 2187*x)*sqrt(-4*x^2 + 9) + 6561*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 = 1.99993, size = 39, normalized size = 0.87 \[ - \frac{x^{3} \sqrt{- 4 x^{2} + 9}}{16} - \frac{27 x \sqrt{- 4 x^{2} + 9}}{128} + \frac{243 \operatorname{asin}{\left (\frac{2 x}{3} \right )}}{256} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(-4*x**2+9)**(1/2),x)

[Out]

-x**3*sqrt(-4*x**2 + 9)/16 - 27*x*sqrt(-4*x**2 + 9)/128 + 243*asin(2*x/3)/256

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GIAC/XCAS [A]  time = 0.21582, size = 35, normalized size = 0.78 \[ -\frac{1}{128} \,{\left (8 \, x^{2} + 27\right )} \sqrt{-4 \, x^{2} + 9} x + \frac{243}{256} \, \arcsin \left (\frac{2}{3} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/sqrt(-4*x^2 + 9),x, algorithm="giac")

[Out]

-1/128*(8*x^2 + 27)*sqrt(-4*x^2 + 9)*x + 243/256*arcsin(2/3*x)