3.470 \(\int \frac{1}{x^4 \left (-8+x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=47 \[ -\frac{x}{192 \sqrt{x^2-8}}+\frac{1}{48 \sqrt{x^2-8} x}+\frac{1}{24 \sqrt{x^2-8} x^3} \]

[Out]

1/(24*x^3*Sqrt[-8 + x^2]) + 1/(48*x*Sqrt[-8 + x^2]) - x/(192*Sqrt[-8 + x^2])

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Rubi [A]  time = 0.0279512, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{x}{192 \sqrt{x^2-8}}+\frac{1}{48 \sqrt{x^2-8} x}+\frac{1}{24 \sqrt{x^2-8} x^3} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*(-8 + x^2)^(3/2)),x]

[Out]

1/(24*x^3*Sqrt[-8 + x^2]) + 1/(48*x*Sqrt[-8 + x^2]) - x/(192*Sqrt[-8 + x^2])

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Rubi in Sympy [A]  time = 2.01458, size = 39, normalized size = 0.83 \[ - \frac{x}{192 \sqrt{x^{2} - 8}} + \frac{1}{48 x \sqrt{x^{2} - 8}} + \frac{1}{24 x^{3} \sqrt{x^{2} - 8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**4/(x**2-8)**(3/2),x)

[Out]

-x/(192*sqrt(x**2 - 8)) + 1/(48*x*sqrt(x**2 - 8)) + 1/(24*x**3*sqrt(x**2 - 8))

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Mathematica [A]  time = 0.0151381, size = 28, normalized size = 0.6 \[ \frac{-x^4+4 x^2+8}{192 x^3 \sqrt{x^2-8}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^4*(-8 + x^2)^(3/2)),x]

[Out]

(8 + 4*x^2 - x^4)/(192*x^3*Sqrt[-8 + x^2])

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Maple [A]  time = 0.005, size = 23, normalized size = 0.5 \[ -{\frac{{x}^{4}-4\,{x}^{2}-8}{192\,{x}^{3}}{\frac{1}{\sqrt{{x}^{2}-8}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(x^2-8)^(3/2),x)

[Out]

-1/192*(x^4-4*x^2-8)/x^3/(x^2-8)^(1/2)

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Maxima [A]  time = 1.5198, size = 47, normalized size = 1. \[ -\frac{x}{192 \, \sqrt{x^{2} - 8}} + \frac{1}{48 \, \sqrt{x^{2} - 8} x} + \frac{1}{24 \, \sqrt{x^{2} - 8} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 - 8)^(3/2)*x^4),x, algorithm="maxima")

[Out]

-1/192*x/sqrt(x^2 - 8) + 1/48/(sqrt(x^2 - 8)*x) + 1/24/(sqrt(x^2 - 8)*x^3)

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Fricas [A]  time = 0.203978, size = 76, normalized size = 1.62 \[ \frac{x^{2} - \sqrt{x^{2} - 8} x - 2}{6 \,{\left (x^{8} - 12 \, x^{6} + 32 \, x^{4} -{\left (x^{7} - 8 \, x^{5} + 8 \, x^{3}\right )} \sqrt{x^{2} - 8}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 - 8)^(3/2)*x^4),x, algorithm="fricas")

[Out]

1/6*(x^2 - sqrt(x^2 - 8)*x - 2)/(x^8 - 12*x^6 + 32*x^4 - (x^7 - 8*x^5 + 8*x^3)*s
qrt(x^2 - 8))

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Sympy [A]  time = 9.77293, size = 153, normalized size = 3.26 \[ \begin{cases} - \frac{i x^{4} \sqrt{-1 + \frac{8}{x^{2}}}}{192 x^{4} - 1536 x^{2}} + \frac{4 i x^{2} \sqrt{-1 + \frac{8}{x^{2}}}}{192 x^{4} - 1536 x^{2}} + \frac{8 i \sqrt{-1 + \frac{8}{x^{2}}}}{192 x^{4} - 1536 x^{2}} & \text{for}\: 8 \left |{\frac{1}{x^{2}}}\right | > 1 \\- \frac{x^{4} \sqrt{1 - \frac{8}{x^{2}}}}{192 x^{4} - 1536 x^{2}} + \frac{4 x^{2} \sqrt{1 - \frac{8}{x^{2}}}}{192 x^{4} - 1536 x^{2}} + \frac{8 \sqrt{1 - \frac{8}{x^{2}}}}{192 x^{4} - 1536 x^{2}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**4/(x**2-8)**(3/2),x)

[Out]

Piecewise((-I*x**4*sqrt(-1 + 8/x**2)/(192*x**4 - 1536*x**2) + 4*I*x**2*sqrt(-1 +
8/x**2)/(192*x**4 - 1536*x**2) + 8*I*sqrt(-1 + 8/x**2)/(192*x**4 - 1536*x**2),
8*Abs(x**(-2)) > 1), (-x**4*sqrt(1 - 8/x**2)/(192*x**4 - 1536*x**2) + 4*x**2*sqr
t(1 - 8/x**2)/(192*x**4 - 1536*x**2) + 8*sqrt(1 - 8/x**2)/(192*x**4 - 1536*x**2)
, True))

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GIAC/XCAS [A]  time = 0.23372, size = 84, normalized size = 1.79 \[ -\frac{x}{512 \, \sqrt{x^{2} - 8}} - \frac{3 \,{\left (x - \sqrt{x^{2} - 8}\right )}^{4} + 96 \,{\left (x - \sqrt{x^{2} - 8}\right )}^{2} + 320}{96 \,{\left ({\left (x - \sqrt{x^{2} - 8}\right )}^{2} + 8\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 - 8)^(3/2)*x^4),x, algorithm="giac")

[Out]

-1/512*x/sqrt(x^2 - 8) - 1/96*(3*(x - sqrt(x^2 - 8))^4 + 96*(x - sqrt(x^2 - 8))^
2 + 320)/((x - sqrt(x^2 - 8))^2 + 8)^3