3.6.63 \(\int \frac {1}{x^4 \sqrt {-9+4 x^2}} \, dx\) [563]

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

[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 = 0.00, 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 = {277, 270} \begin {gather*} \frac {8 \sqrt {4 x^2-9}}{243 x}+\frac {\sqrt {4 x^2-9}}{27 x^3} \end {gather*}

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 270

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 277

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 = 0.03, size = 25, normalized size = 0.68 \begin {gather*} \frac {\sqrt {-9+4 x^2} \left (9+8 x^2\right )}{243 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*Sqrt[-9 + 4*x^2]),x]

[Out]

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

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Maple [A]
time = 0.06, size = 30, normalized size = 0.81

method	result	size
trager	\(\frac {\left (8 x^{2}+9\right ) \sqrt {4 x^{2}-9}}{243 x^{3}}\)	\(22\)
risch	\(\frac {32 x^{4}-36 x^{2}-81}{243 x^{3} \sqrt {4 x^{2}-9}}\)	\(27\)
default	\(\frac {\sqrt {4 x^{2}-9}}{27 x^{3}}+\frac {8 \sqrt {4 x^{2}-9}}{243 x}\)	\(30\)
gosper	\(\frac {\left (2 x -3\right ) \left (2 x +3\right ) \left (8 x^{2}+9\right )}{243 x^{3} \sqrt {4 x^{2}-9}}\)	\(32\)
meijerg	\(-\frac {\sqrt {-\mathrm {signum}\left (-1+\frac {4 x^{2}}{9}\right )}\, \left (1+\frac {8 x^{2}}{9}\right ) \sqrt {1-\frac {4 x^{2}}{9}}}{9 \sqrt {\mathrm {signum}\left (-1+\frac {4 x^{2}}{9}\right )}\, x^{3}}\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.49, size = 29, normalized size = 0.78 \begin {gather*} \frac {8 \, \sqrt {4 \, x^{2} - 9}}{243 \, x} + \frac {\sqrt {4 \, x^{2} - 9}}{27 \, x^{3}} \end {gather*}

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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Fricas [A]
time = 1.59, size = 28, normalized size = 0.76 \begin {gather*} \frac {16 \, x^{3} + {\left (8 \, x^{2} + 9\right )} \sqrt {4 \, x^{2} - 9}}{243 \, x^{3}} \end {gather*}

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

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Sympy [C] Result contains complex when optimal does not.
time = 0.63, size = 68, normalized size = 1.84 \begin {gather*} \begin {cases} \frac {8 \sqrt {4 x^{2} - 9}}{243 x} + \frac {\sqrt {4 x^{2} - 9}}{27 x^{3}} & \text {for}\: \left |{x^{2}}\right | > \frac {9}{4} \\\frac {8 i \sqrt {9 - 4 x^{2}}}{243 x} + \frac {i \sqrt {9 - 4 x^{2}}}{27 x^{3}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((8*sqrt(4*x**2 - 9)/(243*x) + sqrt(4*x**2 - 9)/(27*x**3), Abs(x**2) > 9/4), (8*I*sqrt(9 - 4*x**2)/(2
43*x) + I*sqrt(9 - 4*x**2)/(27*x**3), True))

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Giac [A]
time = 0.89, size = 42, normalized size = 1.14 \begin {gather*} \frac {32 \, {\left ({\left (2 \, x - \sqrt {4 \, x^{2} - 9}\right )}^{2} + 3\right )}}{{\left ({\left (2 \, x - \sqrt {4 \, x^{2} - 9}\right )}^{2} + 9\right )}^{3}} \end {gather*}

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]

32*((2*x - sqrt(4*x^2 - 9))^2 + 3)/((2*x - sqrt(4*x^2 - 9))^2 + 9)^3

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Mupad [B]
time = 4.87, size = 31, normalized size = 0.84 \begin {gather*} \frac {8\,x^2\,\sqrt {4\,x^2-9}+9\,\sqrt {4\,x^2-9}}{243\,x^3} \end {gather*}

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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