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

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

[Out]

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

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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} \[ -\frac {8 \sqrt {9-4 x^2}}{243 x}-\frac {\sqrt {9-4 x^2}}{27 x^3} \]

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 = 0.00, size = 27, normalized size = 0.73 \[ -\frac {\sqrt {1-\frac {4 x^2}{9}} \left (8 x^2+9\right )}{81 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*Sqrt[9 - 4*x^2]),x]

[Out]

-1/81*(Sqrt[1 - (4*x^2)/9]*(9 + 8*x^2))/x^3

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fricas [A]  time = 0.92, 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 [B]  time = 1.13, size = 73, normalized size = 1.97 \[ \frac {2 \, x^{3} {\left (\frac {9 \, {\left (\sqrt {-4 \, x^{2} + 9} - 3\right )}^{2}}{x^{2}} + 4\right )}}{243 \, {\left (\sqrt {-4 \, x^{2} + 9} - 3\right )}^{3}} - \frac {\sqrt {-4 \, x^{2} + 9} - 3}{54 \, x} - \frac {{\left (\sqrt {-4 \, x^{2} + 9} - 3\right )}^{3}}{1944 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/243*x^3*(9*(sqrt(-4*x^2 + 9) - 3)^2/x^2 + 4)/(sqrt(-4*x^2 + 9) - 3)^3 - 1/54*(sqrt(-4*x^2 + 9) - 3)/x - 1/19
44*(sqrt(-4*x^2 + 9) - 3)^3/x^3

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maple [A]  time = 0.00, size = 32, normalized size = 0.86 \[ \frac {\left (2 x -3\right ) \left (2 x +3\right ) \left (8 x^{2}+9\right )}{243 \sqrt {-4 x^{2}+9}\, x^{3}} \]

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

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maxima [A]  time = 2.97, 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 = 0.02, size = 22, normalized size = 0.59 \[ -\sqrt {\frac {9}{4}-x^2}\,\left (\frac {16}{243\,x}+\frac {2}{27\,x^3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(9 - 4*x^2)^(1/2)),x)

[Out]

-(9/4 - x^2)^(1/2)*(16/(243*x) + 2/(27*x^3))

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sympy [A]  time = 1.33, size = 80, normalized size = 2.16 \[ \begin {cases} - \frac {16 \sqrt {-1 + \frac {9}{4 x^{2}}}}{243} - \frac {2 \sqrt {-1 + \frac {9}{4 x^{2}}}}{27 x^{2}} & \text {for}\: \frac {9}{4 \left |{x^{2}}\right |} > 1 \\- \frac {16 i \sqrt {1 - \frac {9}{4 x^{2}}}}{243} - \frac {2 i \sqrt {1 - \frac {9}{4 x^{2}}}}{27 x^{2}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-16*sqrt(-1 + 9/(4*x**2))/243 - 2*sqrt(-1 + 9/(4*x**2))/(27*x**2), 9/(4*Abs(x**2)) > 1), (-16*I*sqr
t(1 - 9/(4*x**2))/243 - 2*I*sqrt(1 - 9/(4*x**2))/(27*x**2), True))

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