3.565 \(\int \frac{x^5}{\sqrt{-9-4 x^2}} \, dx\)

Optimal. Leaf size=46 \[ -\frac{1}{320} \left (-4 x^2-9\right )^{5/2}-\frac{3}{32} \left (-4 x^2-9\right )^{3/2}-\frac{81}{64} \sqrt{-4 x^2-9} \]

[Out]

(-81*Sqrt[-9 - 4*x^2])/64 - (3*(-9 - 4*x^2)^(3/2))/32 - (-9 - 4*x^2)^(5/2)/320

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Rubi [A]  time = 0.0589636, antiderivative size = 46, 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 \[ -\frac{1}{320} \left (-4 x^2-9\right )^{5/2}-\frac{3}{32} \left (-4 x^2-9\right )^{3/2}-\frac{81}{64} \sqrt{-4 x^2-9} \]

Antiderivative was successfully verified.

[In]  Int[x^5/Sqrt[-9 - 4*x^2],x]

[Out]

(-81*Sqrt[-9 - 4*x^2])/64 - (3*(-9 - 4*x^2)^(3/2))/32 - (-9 - 4*x^2)^(5/2)/320

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Rubi in Sympy [A]  time = 6.76829, size = 44, normalized size = 0.96 \[ - \frac{\left (- 4 x^{2} - 9\right )^{\frac{5}{2}}}{320} - \frac{3 \left (- 4 x^{2} - 9\right )^{\frac{3}{2}}}{32} - \frac{81 \sqrt{- 4 x^{2} - 9}}{64} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(-4*x**2-9)**(1/2),x)

[Out]

-(-4*x**2 - 9)**(5/2)/320 - 3*(-4*x**2 - 9)**(3/2)/32 - 81*sqrt(-4*x**2 - 9)/64

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Mathematica [A]  time = 0.0143208, size = 27, normalized size = 0.59 \[ -\frac{1}{40} \sqrt{-4 x^2-9} \left (2 x^4-6 x^2+27\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/Sqrt[-9 - 4*x^2],x]

[Out]

-(Sqrt[-9 - 4*x^2]*(27 - 6*x^2 + 2*x^4))/40

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Maple [A]  time = 0.004, size = 24, normalized size = 0.5 \[ -{\frac{2\,{x}^{4}-6\,{x}^{2}+27}{40}\sqrt{-4\,{x}^{2}-9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(-4*x^2-9)^(1/2),x)

[Out]

-1/40*(2*x^4-6*x^2+27)*(-4*x^2-9)^(1/2)

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Maxima [A]  time = 1.4862, size = 54, normalized size = 1.17 \[ -\frac{1}{20} \, \sqrt{-4 \, x^{2} - 9} x^{4} + \frac{3}{20} \, \sqrt{-4 \, x^{2} - 9} x^{2} - \frac{27}{40} \, \sqrt{-4 \, x^{2} - 9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/sqrt(-4*x^2 - 9),x, algorithm="maxima")

[Out]

-1/20*sqrt(-4*x^2 - 9)*x^4 + 3/20*sqrt(-4*x^2 - 9)*x^2 - 27/40*sqrt(-4*x^2 - 9)

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Fricas [A]  time = 0.21638, size = 31, normalized size = 0.67 \[ -\frac{1}{40} \,{\left (2 \, x^{4} - 6 \, x^{2} + 27\right )} \sqrt{-4 \, x^{2} - 9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/sqrt(-4*x^2 - 9),x, algorithm="fricas")

[Out]

-1/40*(2*x^4 - 6*x^2 + 27)*sqrt(-4*x^2 - 9)

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Sympy [A]  time = 3.54761, size = 49, normalized size = 1.07 \[ - \frac{x^{4} \sqrt{- 4 x^{2} - 9}}{20} + \frac{3 x^{2} \sqrt{- 4 x^{2} - 9}}{20} - \frac{27 \sqrt{- 4 x^{2} - 9}}{40} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(-4*x**2-9)**(1/2),x)

[Out]

-x**4*sqrt(-4*x**2 - 9)/20 + 3*x**2*sqrt(-4*x**2 - 9)/20 - 27*sqrt(-4*x**2 - 9)/
40

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GIAC/XCAS [A]  time = 0.219703, size = 50, normalized size = 1.09 \[ -\frac{1}{320} \,{\left (4 \, x^{2} + 9\right )}^{\frac{5}{2}} i + \frac{3}{32} \,{\left (4 \, x^{2} + 9\right )}^{\frac{3}{2}} i - \frac{81}{64} \, \sqrt{4 \, x^{2} + 9} i \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/sqrt(-4*x^2 - 9),x, algorithm="giac")

[Out]

-1/320*(4*x^2 + 9)^(5/2)*i + 3/32*(4*x^2 + 9)^(3/2)*i - 81/64*sqrt(4*x^2 + 9)*i