3.574 \(\int \frac {1}{x^4 \sqrt {-9-4 x^2}} \, dx\)

Optimal. Leaf size=37 \[ \frac {\sqrt {-4 x^2-9}}{27 x^3}-\frac {8 \sqrt {-4 x^2-9}}{243 x} \]

[Out]

1/27*(-4*x^2-9)^(1/2)/x^3-8/243/x*(-4*x^2-9)^(1/2)

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Rubi [A]  time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {271, 264} \[ \frac {\sqrt {-4 x^2-9}}{27 x^3}-\frac {8 \sqrt {-4 x^2-9}}{243 x} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*Sqrt[-9 - 4*x^2]),x]

[Out]

Sqrt[-9 - 4*x^2]/(27*x^3) - (8*Sqrt[-9 - 4*x^2])/(243*x)

Rule 264

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

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]

Rubi steps

\begin {align*} \int \frac {1}{x^4 \sqrt {-9-4 x^2}} \, dx &=\frac {\sqrt {-9-4 x^2}}{27 x^3}-\frac {8}{27} \int \frac {1}{x^2 \sqrt {-9-4 x^2}} \, dx\\ &=\frac {\sqrt {-9-4 x^2}}{27 x^3}-\frac {8 \sqrt {-9-4 x^2}}{243 x}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 25, normalized size = 0.68 \[ \frac {\left (9-8 x^2\right ) \sqrt {-4 x^2-9}}{243 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*Sqrt[-9 - 4*x^2]),x]

[Out]

((9 - 8*x^2)*Sqrt[-9 - 4*x^2])/(243*x^3)

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fricas [A]  time = 0.57, size = 21, normalized size = 0.57 \[ -\frac {{\left (8 \, x^{2} - 9\right )} \sqrt {-4 \, x^{2} - 9}}{243 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(-4*x^2-9)^(1/2),x, algorithm="fricas")

[Out]

-1/243*(8*x^2 - 9)*sqrt(-4*x^2 - 9)/x^3

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-4 \, x^{2} - 9} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(-4*x^2-9)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-4*x^2 - 9)*x^4), x)

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maple [A]  time = 0.00, size = 22, normalized size = 0.59 \[ -\frac {\left (8 x^{2}-9\right ) \sqrt {-4 x^{2}-9}}{243 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(-4*x^2-9)^(1/2),x)

[Out]

-1/243*(8*x^2-9)/x^3*(-4*x^2-9)^(1/2)

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maxima [A]  time = 2.96, size = 29, normalized size = 0.78 \[ -\frac {8 \, \sqrt {-4 \, x^{2} - 9}}{243 \, x} + \frac {\sqrt {-4 \, x^{2} - 9}}{27 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(-4*x^2-9)^(1/2),x, algorithm="maxima")

[Out]

-8/243*sqrt(-4*x^2 - 9)/x + 1/27*sqrt(-4*x^2 - 9)/x^3

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mupad [B]  time = 5.08, size = 31, normalized size = 0.84 \[ -\frac {8\,x^2\,\sqrt {-4\,x^2-9}-9\,\sqrt {-4\,x^2-9}}{243\,x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(- 4*x^2 - 9)^(1/2)),x)

[Out]

-(8*x^2*(- 4*x^2 - 9)^(1/2) - 9*(- 4*x^2 - 9)^(1/2))/(243*x^3)

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sympy [C]  time = 1.29, size = 36, normalized size = 0.97 \[ - \frac {16 i \sqrt {1 + \frac {9}{4 x^{2}}}}{243} + \frac {2 i \sqrt {1 + \frac {9}{4 x^{2}}}}{27 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-16*I*sqrt(1 + 9/(4*x**2))/243 + 2*I*sqrt(1 + 9/(4*x**2))/(27*x**2)

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