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

Optimal. Leaf size=54 \[ \frac{27}{128} \sqrt{-4 x^2-9} x+\frac{243}{256} \tan ^{-1}\left (\frac{2 x}{\sqrt{-4 x^2-9}}\right )-\frac{1}{16} \sqrt{-4 x^2-9} x^3 \]

[Out]

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

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Rubi [A]  time = 0.0482272, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{27}{128} \sqrt{-4 x^2-9} x+\frac{243}{256} \tan ^{-1}\left (\frac{2 x}{\sqrt{-4 x^2-9}}\right )-\frac{1}{16} \sqrt{-4 x^2-9} x^3 \]

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*ArcTan[(2*x)/Sqrt
[-9 - 4*x^2]])/256

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Rubi in Sympy [A]  time = 5.58185, size = 53, normalized size = 0.98 \[ - \frac{x^{3} \sqrt{- 4 x^{2} - 9}}{16} + \frac{27 x \sqrt{- 4 x^{2} - 9}}{128} + \frac{243 \operatorname{atan}{\left (\frac{2 x}{\sqrt{- 4 x^{2} - 9}} \right )}}{256} \]

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*atan(2*x/sqrt(-4*x
**2 - 9))/256

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Mathematica [A]  time = 0.0382876, size = 43, normalized size = 0.8 \[ \frac{1}{256} \left (2 x \sqrt{-4 x^2-9} \left (27-8 x^2\right )+243 \tan ^{-1}\left (\frac{2 x}{\sqrt{-4 x^2-9}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*x*(27 - 8*x^2)*Sqrt[-9 - 4*x^2] + 243*ArcTan[(2*x)/Sqrt[-9 - 4*x^2]])/256

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Maple [A]  time = 0.009, size = 43, normalized size = 0.8 \[{\frac{243}{256}\arctan \left ( 2\,{\frac{x}{\sqrt{-4\,{x}^{2}-9}}} \right ) }+{\frac{27\,x}{128}\sqrt{-4\,{x}^{2}-9}}-{\frac{{x}^{3}}{16}\sqrt{-4\,{x}^{2}-9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(-4*x^2-9)^(1/2),x)

[Out]

243/256*arctan(2*x/(-4*x^2-9)^(1/2))+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 = 1.50375, size = 45, normalized size = 0.83 \[ -\frac{1}{16} \, \sqrt{-4 \, x^{2} - 9} x^{3} + \frac{27}{128} \, \sqrt{-4 \, x^{2} - 9} x - \frac{243}{256} i \, \operatorname{arsinh}\left (\frac{2}{3} \, x\right ) \]

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*I*arcsinh(2/3*x
)

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Fricas [A]  time = 0.225959, size = 90, normalized size = 1.67 \[ -\frac{1}{128} \,{\left (8 \, x^{3} - 27 \, x\right )} \sqrt{-4 \, x^{2} - 9} + \frac{243}{512} i \, \log \left (-\frac{8 \, x + 4 i \, \sqrt{-4 \, x^{2} - 9}}{x}\right ) - \frac{243}{512} i \, \log \left (-\frac{8 \, x - 4 i \, \sqrt{-4 \, x^{2} - 9}}{x}\right ) \]

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*(8*x^3 - 27*x)*sqrt(-4*x^2 - 9) + 243/512*I*log(-(8*x + 4*I*sqrt(-4*x^2 -
9))/x) - 243/512*I*log(-(8*x - 4*I*sqrt(-4*x^2 - 9))/x)

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Sympy [A]  time = 3.00486, size = 53, normalized size = 0.98 \[ - \frac{x^{3} \sqrt{- 4 x^{2} - 9}}{16} + \frac{27 x \sqrt{- 4 x^{2} - 9}}{128} + \frac{243 \operatorname{atan}{\left (\frac{2 x}{\sqrt{- 4 x^{2} - 9}} \right )}}{256} \]

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*atan(2*x/sqrt(-4*x
**2 - 9))/256

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GIAC/XCAS [A]  time = 0.226112, size = 38, normalized size = 0.7 \[ -\frac{1}{128} \,{\left (8 \, x^{2} - 27\right )} \sqrt{-4 \, x^{2} - 9} x - \frac{243}{256} \, i \arcsin \left (\frac{2}{3} \, i x\right ) \]

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*i*arcsin(2/3*i*x)