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

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 = 0.0094631, 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, Rules used = {271, 191} \[ -\frac{x}{192 \sqrt{x^2-8}}+\frac{1}{48 \sqrt{x^2-8} x}+\frac{1}{24 \sqrt{x^2-8} x^3} \]

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])

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

\begin{align*} \int \frac{1}{x^4 \left (-8+x^2\right )^{3/2}} \, dx &=\frac{1}{24 x^3 \sqrt{-8+x^2}}+\frac{1}{6} \int \frac{1}{x^2 \left (-8+x^2\right )^{3/2}} \, dx\\ &=\frac{1}{24 x^3 \sqrt{-8+x^2}}+\frac{1}{48 x \sqrt{-8+x^2}}+\frac{1}{24} \int \frac{1}{\left (-8+x^2\right )^{3/2}} \, dx\\ &=\frac{1}{24 x^3 \sqrt{-8+x^2}}+\frac{1}{48 x \sqrt{-8+x^2}}-\frac{x}{192 \sqrt{-8+x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0052976, 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 = 0.002, size = 23, normalized size = 0.5 \begin{align*} -{\frac{{x}^{4}-4\,{x}^{2}-8}{192\,{x}^{3}}{\frac{1}{\sqrt{{x}^{2}-8}}}} \end{align*}

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 = 1.40514, size = 47, normalized size = 1. \begin{align*} -\frac{x}{192 \, \sqrt{x^{2} - 8}} + \frac{1}{48 \, \sqrt{x^{2} - 8} x} + \frac{1}{24 \, \sqrt{x^{2} - 8} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(x^2-8)^(3/2),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 = 2.04825, size = 95, normalized size = 2.02 \begin{align*} -\frac{x^{5} - 8 \, x^{3} +{\left (x^{4} - 4 \, x^{2} - 8\right )} \sqrt{x^{2} - 8}}{192 \,{\left (x^{5} - 8 \, x^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/192*(x^5 - 8*x^3 + (x^4 - 4*x^2 - 8)*sqrt(x^2 - 8))/(x^5 - 8*x^3)

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Sympy [A]  time = 4.75237, size = 151, normalized size = 3.21 \begin{align*} \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}\: \frac{8}{\left |{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} \end{align*}

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*sqrt(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 [A]  time = 1.07795, size = 84, normalized size = 1.79 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(x^2-8)^(3/2),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