3.6.41 \(\int \frac {1}{x^4 \sqrt {9+4 x^2}} \, dx\) [541]

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 = 0.01, 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 = {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]

-1/27*Sqrt[9 + 4*x^2]/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 = 0.05, size = 30, normalized size = 0.81

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

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 = 0.63, 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 = 1.39, 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 [A]
time = 0.64, size = 32, normalized size = 0.86 \begin {gather*} \frac {16 \sqrt {1 + \frac {9}{4 x^{2}}}}{243} - \frac {2 \sqrt {1 + \frac {9}{4 x^{2}}}}{27 x^{2}} \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]

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

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Giac [A]
time = 1.17, 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 = 0.02, size = 19, normalized size = 0.51 \begin {gather*} \sqrt {x^2+\frac {9}{4}}\,\left (\frac {16}{243\,x}-\frac {2}{27\,x^3}\right ) \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]

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

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