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

Optimal. Leaf size=72 \[ -\frac{81}{256} \sqrt{-4 x^2-9} x-\frac{729}{512} \tan ^{-1}\left (\frac{2 x}{\sqrt{-4 x^2-9}}\right )+\frac{1}{6} \sqrt{-4 x^2-9} x^5+\frac{3}{32} \sqrt{-4 x^2-9} x^3 \]

[Out]

(-81*x*Sqrt[-9 - 4*x^2])/256 + (3*x^3*Sqrt[-9 - 4*x^2])/32 + (x^5*Sqrt[-9 - 4*x^
2])/6 - (729*ArcTan[(2*x)/Sqrt[-9 - 4*x^2]])/512

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Rubi [A]  time = 0.0666989, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{81}{256} \sqrt{-4 x^2-9} x-\frac{729}{512} \tan ^{-1}\left (\frac{2 x}{\sqrt{-4 x^2-9}}\right )+\frac{1}{6} \sqrt{-4 x^2-9} x^5+\frac{3}{32} \sqrt{-4 x^2-9} x^3 \]

Antiderivative was successfully verified.

[In]  Int[x^4*Sqrt[-9 - 4*x^2],x]

[Out]

(-81*x*Sqrt[-9 - 4*x^2])/256 + (3*x^3*Sqrt[-9 - 4*x^2])/32 + (x^5*Sqrt[-9 - 4*x^
2])/6 - (729*ArcTan[(2*x)/Sqrt[-9 - 4*x^2]])/512

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Rubi in Sympy [A]  time = 7.78677, size = 71, normalized size = 0.99 \[ \frac{x^{5} \sqrt{- 4 x^{2} - 9}}{6} + \frac{3 x^{3} \sqrt{- 4 x^{2} - 9}}{32} - \frac{81 x \sqrt{- 4 x^{2} - 9}}{256} - \frac{729 \operatorname{atan}{\left (\frac{2 x}{\sqrt{- 4 x^{2} - 9}} \right )}}{512} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**5*sqrt(-4*x**2 - 9)/6 + 3*x**3*sqrt(-4*x**2 - 9)/32 - 81*x*sqrt(-4*x**2 - 9)/
256 - 729*atan(2*x/sqrt(-4*x**2 - 9))/512

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Mathematica [A]  time = 0.0302694, size = 48, normalized size = 0.67 \[ \frac{1}{768} x \sqrt{-4 x^2-9} \left (128 x^4+72 x^2-243\right )-\frac{729}{512} \tan ^{-1}\left (\frac{2 x}{\sqrt{-4 x^2-9}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[-9 - 4*x^2]*(-243 + 72*x^2 + 128*x^4))/768 - (729*ArcTan[(2*x)/Sqrt[-9 -
4*x^2]])/512

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Maple [A]  time = 0.012, size = 55, normalized size = 0.8 \[ -{\frac{{x}^{3}}{24} \left ( -4\,{x}^{2}-9 \right ) ^{{\frac{3}{2}}}}+{\frac{9\,x}{128} \left ( -4\,{x}^{2}-9 \right ) ^{{\frac{3}{2}}}}+{\frac{81\,x}{256}\sqrt{-4\,{x}^{2}-9}}-{\frac{729}{512}\arctan \left ( 2\,{\frac{x}{\sqrt{-4\,{x}^{2}-9}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(-4*x^2-9)^(1/2),x)

[Out]

-1/24*x^3*(-4*x^2-9)^(3/2)+9/128*x*(-4*x^2-9)^(3/2)+81/256*x*(-4*x^2-9)^(1/2)-72
9/512*arctan(2*x/(-4*x^2-9)^(1/2))

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Maxima [A]  time = 1.50604, size = 61, normalized size = 0.85 \[ -\frac{1}{24} \,{\left (-4 \, x^{2} - 9\right )}^{\frac{3}{2}} x^{3} + \frac{9}{128} \,{\left (-4 \, x^{2} - 9\right )}^{\frac{3}{2}} x + \frac{81}{256} \, \sqrt{-4 \, x^{2} - 9} x + \frac{729}{512} i \, \operatorname{arsinh}\left (\frac{2}{3} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/24*(-4*x^2 - 9)^(3/2)*x^3 + 9/128*(-4*x^2 - 9)^(3/2)*x + 81/256*sqrt(-4*x^2 -
9)*x + 729/512*I*arcsinh(2/3*x)

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Fricas [A]  time = 0.235006, size = 97, normalized size = 1.35 \[ \frac{1}{768} \,{\left (128 \, x^{5} + 72 \, x^{3} - 243 \, x\right )} \sqrt{-4 \, x^{2} - 9} - \frac{729}{1024} i \, \log \left (-\frac{8 \, x + 4 i \, \sqrt{-4 \, x^{2} - 9}}{x}\right ) + \frac{729}{1024} i \, \log \left (-\frac{8 \, x - 4 i \, \sqrt{-4 \, x^{2} - 9}}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/768*(128*x^5 + 72*x^3 - 243*x)*sqrt(-4*x^2 - 9) - 729/1024*I*log(-(8*x + 4*I*s
qrt(-4*x^2 - 9))/x) + 729/1024*I*log(-(8*x - 4*I*sqrt(-4*x^2 - 9))/x)

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Sympy [A]  time = 15.3896, size = 83, normalized size = 1.15 \[ \frac{2 i x^{7}}{3 \sqrt{4 x^{2} + 9}} + \frac{15 i x^{5}}{8 \sqrt{4 x^{2} + 9}} - \frac{27 i x^{3}}{64 \sqrt{4 x^{2} + 9}} - \frac{729 i x}{256 \sqrt{4 x^{2} + 9}} + \frac{729 i \operatorname{asinh}{\left (\frac{2 x}{3} \right )}}{512} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*I*x**7/(3*sqrt(4*x**2 + 9)) + 15*I*x**5/(8*sqrt(4*x**2 + 9)) - 27*I*x**3/(64*s
qrt(4*x**2 + 9)) - 729*I*x/(256*sqrt(4*x**2 + 9)) + 729*I*asinh(2*x/3)/512

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GIAC/XCAS [A]  time = 0.207161, size = 47, normalized size = 0.65 \[ \frac{1}{768} \,{\left (8 \,{\left (16 \, x^{2} + 9\right )} x^{2} - 243\right )} \sqrt{-4 \, x^{2} - 9} x + \frac{729}{512} \, i \arcsin \left (\frac{2}{3} \, i x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/768*(8*(16*x^2 + 9)*x^2 - 243)*sqrt(-4*x^2 - 9)*x + 729/512*i*arcsin(2/3*i*x)