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

Optimal. Leaf size=18 \[ -\frac {\left (9-4 x^2\right )^{3/2}}{27 x^3} \]

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {264} \[ -\frac {\left (9-4 x^2\right )^{3/2}}{27 x^3} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[9 - 4*x^2]/x^4,x]

[Out]

-(9 - 4*x^2)^(3/2)/(27*x^3)

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]

Rubi steps

\begin {align*} \int \frac {\sqrt {9-4 x^2}}{x^4} \, dx &=-\frac {\left (9-4 x^2\right )^{3/2}}{27 x^3}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 18, normalized size = 1.00 \[ -\frac {\left (9-4 x^2\right )^{3/2}}{27 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[9 - 4*x^2]/x^4,x]

[Out]

-1/27*(9 - 4*x^2)^(3/2)/x^3

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fricas [A]  time = 0.91, size = 21, normalized size = 1.17 \[ \frac {{\left (4 \, x^{2} - 9\right )} \sqrt {-4 \, x^{2} + 9}}{27 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/27*(4*x^2 - 9)*sqrt(-4*x^2 + 9)/x^3

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giac [B]  time = 1.05, size = 73, normalized size = 4.06 \[ -\frac {2 \, x^{3} {\left (\frac {3 \, {\left (\sqrt {-4 \, x^{2} + 9} - 3\right )}^{2}}{x^{2}} - 4\right )}}{27 \, {\left (\sqrt {-4 \, x^{2} + 9} - 3\right )}^{3}} + \frac {\sqrt {-4 \, x^{2} + 9} - 3}{18 \, x} - \frac {{\left (\sqrt {-4 \, x^{2} + 9} - 3\right )}^{3}}{216 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/27*x^3*(3*(sqrt(-4*x^2 + 9) - 3)^2/x^2 - 4)/(sqrt(-4*x^2 + 9) - 3)^3 + 1/18*(sqrt(-4*x^2 + 9) - 3)/x - 1/21
6*(sqrt(-4*x^2 + 9) - 3)^3/x^3

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maple [A]  time = 0.00, size = 25, normalized size = 1.39 \[ \frac {\left (2 x -3\right ) \left (2 x +3\right ) \sqrt {-4 x^{2}+9}}{27 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.90, size = 14, normalized size = 0.78 \[ -\frac {{\left (-4 \, x^{2} + 9\right )}^{\frac {3}{2}}}{27 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/27*(-4*x^2 + 9)^(3/2)/x^3

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mupad [B]  time = 0.03, size = 31, normalized size = 1.72 \[ \frac {8\,x^2\,\sqrt {\frac {9}{4}-x^2}-18\,\sqrt {\frac {9}{4}-x^2}}{27\,x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((9 - 4*x^2)^(1/2)/x^4,x)

[Out]

(8*x^2*(9/4 - x^2)^(1/2) - 18*(9/4 - x^2)^(1/2))/(27*x^3)

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sympy [B]  time = 0.97, size = 76, normalized size = 4.22 \[ \begin {cases} \frac {8 \sqrt {-1 + \frac {9}{4 x^{2}}}}{27} - \frac {2 \sqrt {-1 + \frac {9}{4 x^{2}}}}{3 x^{2}} & \text {for}\: \frac {9}{4 \left |{x^{2}}\right |} > 1 \\\frac {8 i \sqrt {1 - \frac {9}{4 x^{2}}}}{27} - \frac {2 i \sqrt {1 - \frac {9}{4 x^{2}}}}{3 x^{2}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((8*sqrt(-1 + 9/(4*x**2))/27 - 2*sqrt(-1 + 9/(4*x**2))/(3*x**2), 9/(4*Abs(x**2)) > 1), (8*I*sqrt(1 -
9/(4*x**2))/27 - 2*I*sqrt(1 - 9/(4*x**2))/(3*x**2), True))

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