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

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

[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} \[ \frac {\left (4 x^2-9\right )^{3/2}}{27 x^3} \]

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 \[ \frac {\left (4 x^2-9\right )^{3/2}}{27 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-9 + 4*x^2]/x^4,x]

[Out]

(-9 + 4*x^2)^(3/2)/(27*x^3)

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fricas [A]  time = 0.90, size = 20, normalized size = 1.11 \[ \frac {8 \, x^{3} + {\left (4 \, x^{2} - 9\right )}^{\frac {3}{2}}}{27 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/27*(8*x^3 + (4*x^2 - 9)^(3/2))/x^3

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giac [B]  time = 1.10, size = 42, normalized size = 2.33 \[ \frac {16 \, {\left ({\left (2 \, x - \sqrt {4 \, x^{2} - 9}\right )}^{4} + 27\right )}}{{\left ({\left (2 \, x - \sqrt {4 \, x^{2} - 9}\right )}^{2} + 9\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

16*((2*x - sqrt(4*x^2 - 9))^4 + 27)/((2*x - sqrt(4*x^2 - 9))^2 + 9)^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 = 3.01, 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 = 5.10, size = 31, normalized size = 1.72 \[ \frac {4\,x^2\,\sqrt {4\,x^2-9}-9\,\sqrt {4\,x^2-9}}{27\,x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^2 - 9)^(1/2)/x^4,x)

[Out]

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

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sympy [B]  time = 0.98, size = 76, normalized size = 4.22 \[ \begin {cases} \frac {8 i \sqrt {-1 + \frac {9}{4 x^{2}}}}{27} - \frac {2 i \sqrt {-1 + \frac {9}{4 x^{2}}}}{3 x^{2}} & \text {for}\: \frac {9}{4 \left |{x^{2}}\right |} > 1 \\\frac {8 \sqrt {1 - \frac {9}{4 x^{2}}}}{27} - \frac {2 \sqrt {1 - \frac {9}{4 x^{2}}}}{3 x^{2}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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