3.462 \(\int \frac{\sqrt{9-4 x^2}}{x^5} \, dx\)

Optimal. Leaf size=57 \[ \frac{\sqrt{9-4 x^2}}{18 x^2}-\frac{\sqrt{9-4 x^2}}{4 x^4}+\frac{2}{27} \tanh ^{-1}\left (\frac{1}{3} \sqrt{9-4 x^2}\right ) \]

[Out]

-Sqrt[9 - 4*x^2]/(4*x^4) + Sqrt[9 - 4*x^2]/(18*x^2) + (2*ArcTanh[Sqrt[9 - 4*x^2]/3])/27

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Rubi [A]  time = 0.0219855, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {266, 47, 51, 63, 206} \[ \frac{\sqrt{9-4 x^2}}{18 x^2}-\frac{\sqrt{9-4 x^2}}{4 x^4}+\frac{2}{27} \tanh ^{-1}\left (\frac{1}{3} \sqrt{9-4 x^2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[9 - 4*x^2]/x^5,x]

[Out]

-Sqrt[9 - 4*x^2]/(4*x^4) + Sqrt[9 - 4*x^2]/(18*x^2) + (2*ArcTanh[Sqrt[9 - 4*x^2]/3])/27

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\sqrt{9-4 x^2}}{x^5} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\sqrt{9-4 x}}{x^3} \, dx,x,x^2\right )\\ &=-\frac{\sqrt{9-4 x^2}}{4 x^4}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{9-4 x} x^2} \, dx,x,x^2\right )\\ &=-\frac{\sqrt{9-4 x^2}}{4 x^4}+\frac{\sqrt{9-4 x^2}}{18 x^2}-\frac{1}{9} \operatorname{Subst}\left (\int \frac{1}{\sqrt{9-4 x} x} \, dx,x,x^2\right )\\ &=-\frac{\sqrt{9-4 x^2}}{4 x^4}+\frac{\sqrt{9-4 x^2}}{18 x^2}+\frac{1}{18} \operatorname{Subst}\left (\int \frac{1}{\frac{9}{4}-\frac{x^2}{4}} \, dx,x,\sqrt{9-4 x^2}\right )\\ &=-\frac{\sqrt{9-4 x^2}}{4 x^4}+\frac{\sqrt{9-4 x^2}}{18 x^2}+\frac{2}{27} \tanh ^{-1}\left (\frac{1}{3} \sqrt{9-4 x^2}\right )\\ \end{align*}

Mathematica [C]  time = 0.005111, size = 32, normalized size = 0.56 \[ -\frac{16 \left (9-4 x^2\right )^{3/2} \, _2F_1\left (\frac{3}{2},3;\frac{5}{2};1-\frac{4 x^2}{9}\right )}{2187} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[9 - 4*x^2]/x^5,x]

[Out]

(-16*(9 - 4*x^2)^(3/2)*Hypergeometric2F1[3/2, 3, 5/2, 1 - (4*x^2)/9])/2187

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Maple [A]  time = 0.004, size = 55, normalized size = 1. \begin{align*} -{\frac{1}{36\,{x}^{4}} \left ( -4\,{x}^{2}+9 \right ) ^{{\frac{3}{2}}}}-{\frac{1}{162\,{x}^{2}} \left ( -4\,{x}^{2}+9 \right ) ^{{\frac{3}{2}}}}-{\frac{2}{81}\sqrt{-4\,{x}^{2}+9}}+{\frac{2}{27}{\it Artanh} \left ( 3\,{\frac{1}{\sqrt{-4\,{x}^{2}+9}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*x^2+9)^(1/2)/x^5,x)

[Out]

-1/36/x^4*(-4*x^2+9)^(3/2)-1/162/x^2*(-4*x^2+9)^(3/2)-2/81*(-4*x^2+9)^(1/2)+2/27*arctanh(3/(-4*x^2+9)^(1/2))

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Maxima [A]  time = 2.87239, size = 88, normalized size = 1.54 \begin{align*} -\frac{2}{81} \, \sqrt{-4 \, x^{2} + 9} - \frac{{\left (-4 \, x^{2} + 9\right )}^{\frac{3}{2}}}{162 \, x^{2}} - \frac{{\left (-4 \, x^{2} + 9\right )}^{\frac{3}{2}}}{36 \, x^{4}} + \frac{2}{27} \, \log \left (\frac{6 \, \sqrt{-4 \, x^{2} + 9}}{{\left | x \right |}} + \frac{18}{{\left | x \right |}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^2+9)^(1/2)/x^5,x, algorithm="maxima")

[Out]

-2/81*sqrt(-4*x^2 + 9) - 1/162*(-4*x^2 + 9)^(3/2)/x^2 - 1/36*(-4*x^2 + 9)^(3/2)/x^4 + 2/27*log(6*sqrt(-4*x^2 +
9)/abs(x) + 18/abs(x))

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Fricas [A]  time = 1.53264, size = 112, normalized size = 1.96 \begin{align*} -\frac{8 \, x^{4} \log \left (\frac{\sqrt{-4 \, x^{2} + 9} - 3}{x}\right ) - 3 \,{\left (2 \, x^{2} - 9\right )} \sqrt{-4 \, x^{2} + 9}}{108 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^2+9)^(1/2)/x^5,x, algorithm="fricas")

[Out]

-1/108*(8*x^4*log((sqrt(-4*x^2 + 9) - 3)/x) - 3*(2*x^2 - 9)*sqrt(-4*x^2 + 9))/x^4

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Sympy [A]  time = 3.52182, size = 139, normalized size = 2.44 \begin{align*} \begin{cases} \frac{2 \operatorname{acosh}{\left (\frac{3}{2 x} \right )}}{27} - \frac{1}{9 x \sqrt{-1 + \frac{9}{4 x^{2}}}} + \frac{3}{4 x^{3} \sqrt{-1 + \frac{9}{4 x^{2}}}} - \frac{9}{8 x^{5} \sqrt{-1 + \frac{9}{4 x^{2}}}} & \text{for}\: \frac{9}{4 \left |{x^{2}}\right |} > 1 \\- \frac{2 i \operatorname{asin}{\left (\frac{3}{2 x} \right )}}{27} + \frac{i}{9 x \sqrt{1 - \frac{9}{4 x^{2}}}} - \frac{3 i}{4 x^{3} \sqrt{1 - \frac{9}{4 x^{2}}}} + \frac{9 i}{8 x^{5} \sqrt{1 - \frac{9}{4 x^{2}}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x**2+9)**(1/2)/x**5,x)

[Out]

Piecewise((2*acosh(3/(2*x))/27 - 1/(9*x*sqrt(-1 + 9/(4*x**2))) + 3/(4*x**3*sqrt(-1 + 9/(4*x**2))) - 9/(8*x**5*
sqrt(-1 + 9/(4*x**2))), 9/(4*Abs(x**2)) > 1), (-2*I*asin(3/(2*x))/27 + I/(9*x*sqrt(1 - 9/(4*x**2))) - 3*I/(4*x
**3*sqrt(1 - 9/(4*x**2))) + 9*I/(8*x**5*sqrt(1 - 9/(4*x**2))), True))

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Giac [A]  time = 1.98292, size = 77, normalized size = 1.35 \begin{align*} -\frac{{\left (-4 \, x^{2} + 9\right )}^{\frac{3}{2}} + 9 \, \sqrt{-4 \, x^{2} + 9}}{72 \, x^{4}} + \frac{1}{27} \, \log \left (\sqrt{-4 \, x^{2} + 9} + 3\right ) - \frac{1}{27} \, \log \left (-\sqrt{-4 \, x^{2} + 9} + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^2+9)^(1/2)/x^5,x, algorithm="giac")

[Out]

-1/72*((-4*x^2 + 9)^(3/2) + 9*sqrt(-4*x^2 + 9))/x^4 + 1/27*log(sqrt(-4*x^2 + 9) + 3) - 1/27*log(-sqrt(-4*x^2 +
9) + 3)