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

Optimal. Leaf size=54 \[ \frac{9}{32} \sqrt{-4 x^2-9} x+\frac{81}{64} \tan ^{-1}\left (\frac{2 x}{\sqrt{-4 x^2-9}}\right )+\frac{1}{4} \sqrt{-4 x^2-9} x^3 \]

[Out]

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

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Rubi [A]  time = 0.0469325, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{9}{32} \sqrt{-4 x^2-9} x+\frac{81}{64} \tan ^{-1}\left (\frac{2 x}{\sqrt{-4 x^2-9}}\right )+\frac{1}{4} \sqrt{-4 x^2-9} x^3 \]

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*ArcTan[(2*x)/Sqrt[-9
- 4*x^2]])/64

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Rubi in Sympy [A]  time = 6.12681, size = 53, normalized size = 0.98 \[ \frac{x^{3} \sqrt{- 4 x^{2} - 9}}{4} + \frac{9 x \sqrt{- 4 x^{2} - 9}}{32} + \frac{81 \operatorname{atan}{\left (\frac{2 x}{\sqrt{- 4 x^{2} - 9}} \right )}}{64} \]

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*atan(2*x/sqrt(-4*x**2 -
9))/64

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Mathematica [A]  time = 0.0320073, size = 43, normalized size = 0.8 \[ \frac{1}{64} \left (2 x \sqrt{-4 x^2-9} \left (8 x^2+9\right )+81 \tan ^{-1}\left (\frac{2 x}{\sqrt{-4 x^2-9}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*x*Sqrt[-9 - 4*x^2]*(9 + 8*x^2) + 81*ArcTan[(2*x)/Sqrt[-9 - 4*x^2]])/64

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Maple [A]  time = 0.007, size = 41, normalized size = 0.8 \[ -{\frac{x}{16} \left ( -4\,{x}^{2}-9 \right ) ^{{\frac{3}{2}}}}-{\frac{9\,x}{32}\sqrt{-4\,{x}^{2}-9}}+{\frac{81}{64}\arctan \left ( 2\,{\frac{x}{\sqrt{-4\,{x}^{2}-9}}} \right ) } \]

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*arctan(2*x/(-4*x^2-9)^(1/
2))

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Maxima [A]  time = 1.50434, size = 42, normalized size = 0.78 \[ -\frac{1}{16} \,{\left (-4 \, x^{2} - 9\right )}^{\frac{3}{2}} x - \frac{9}{32} \, \sqrt{-4 \, x^{2} - 9} x - \frac{81}{64} i \, \operatorname{arsinh}\left (\frac{2}{3} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.241599, size = 90, normalized size = 1.67 \[ \frac{1}{32} \,{\left (8 \, x^{3} + 9 \, x\right )} \sqrt{-4 \, x^{2} - 9} + \frac{81}{128} i \, \log \left (-\frac{8 \, x + 4 i \, \sqrt{-4 \, x^{2} - 9}}{x}\right ) - \frac{81}{128} i \, \log \left (-\frac{8 \, x - 4 i \, \sqrt{-4 \, x^{2} - 9}}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/32*(8*x^3 + 9*x)*sqrt(-4*x^2 - 9) + 81/128*I*log(-(8*x + 4*I*sqrt(-4*x^2 - 9))
/x) - 81/128*I*log(-(8*x - 4*I*sqrt(-4*x^2 - 9))/x)

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Sympy [A]  time = 9.06351, size = 61, normalized size = 1.13 \[ \frac{i x^{5}}{\sqrt{4 x^{2} + 9}} + \frac{27 i x^{3}}{8 \sqrt{4 x^{2} + 9}} + \frac{81 i x}{32 \sqrt{4 x^{2} + 9}} - \frac{81 i \operatorname{asinh}{\left (\frac{2 x}{3} \right )}}{64} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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*asinh(2*x/3)/64

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GIAC/XCAS [A]  time = 0.2091, size = 38, normalized size = 0.7 \[ \frac{1}{32} \,{\left (8 \, x^{2} + 9\right )} \sqrt{-4 \, x^{2} - 9} x - \frac{81}{64} \, i \arcsin \left (\frac{2}{3} \, i x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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