3.471 \(\int \frac{\sqrt{-9+4 x^2}}{x^3} \, dx\)

Optimal. Leaf size=39 \[ \frac{2}{3} \tan ^{-1}\left (\frac{1}{3} \sqrt{4 x^2-9}\right )-\frac{\sqrt{4 x^2-9}}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.016156, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 47, 63, 203} \[ \frac{2}{3} \tan ^{-1}\left (\frac{1}{3} \sqrt{4 x^2-9}\right )-\frac{\sqrt{4 x^2-9}}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[-9 + 4*x^2]/x^3,x]

[Out]

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

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 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 203

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

Rubi steps

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

Mathematica [A]  time = 0.0074068, size = 55, normalized size = 1.41 \[ -\frac{12 x^2+4 \sqrt{9-4 x^2} x^2 \tanh ^{-1}\left (\sqrt{1-\frac{4 x^2}{9}}\right )-27}{6 x^2 \sqrt{4 x^2-9}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-9 + 4*x^2]/x^3,x]

[Out]

-(-27 + 12*x^2 + 4*x^2*Sqrt[9 - 4*x^2]*ArcTanh[Sqrt[1 - (4*x^2)/9]])/(6*x^2*Sqrt[-9 + 4*x^2])

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Maple [A]  time = 0.003, size = 41, normalized size = 1.1 \begin{align*}{\frac{1}{18\,{x}^{2}} \left ( 4\,{x}^{2}-9 \right ) ^{{\frac{3}{2}}}}-{\frac{2}{9}\sqrt{4\,{x}^{2}-9}}-{\frac{2}{3}\arctan \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^3,x)

[Out]

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

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Maxima [A]  time = 3.93929, size = 47, normalized size = 1.21 \begin{align*} -\frac{2}{9} \, \sqrt{4 \, x^{2} - 9} + \frac{{\left (4 \, x^{2} - 9\right )}^{\frac{3}{2}}}{18 \, x^{2}} - \frac{2}{3} \, \arcsin \left (\frac{3}{2 \,{\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^3,x, algorithm="maxima")

[Out]

-2/9*sqrt(4*x^2 - 9) + 1/18*(4*x^2 - 9)^(3/2)/x^2 - 2/3*arcsin(3/2/abs(x))

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Fricas [A]  time = 1.56207, size = 100, normalized size = 2.56 \begin{align*} \frac{8 \, x^{2} \arctan \left (-\frac{2}{3} \, x + \frac{1}{3} \, \sqrt{4 \, x^{2} - 9}\right ) - 3 \, \sqrt{4 \, x^{2} - 9}}{6 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2-9)^(1/2)/x^3,x, algorithm="fricas")

[Out]

1/6*(8*x^2*arctan(-2/3*x + 1/3*sqrt(4*x^2 - 9)) - 3*sqrt(4*x^2 - 9))/x^2

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Sympy [A]  time = 1.78121, size = 97, normalized size = 2.49 \begin{align*} \begin{cases} \frac{2 i \operatorname{acosh}{\left (\frac{3}{2 x} \right )}}{3} + \frac{i}{x \sqrt{-1 + \frac{9}{4 x^{2}}}} - \frac{9 i}{4 x^{3} \sqrt{-1 + \frac{9}{4 x^{2}}}} & \text{for}\: \frac{9}{4 \left |{x^{2}}\right |} > 1 \\- \frac{2 \operatorname{asin}{\left (\frac{3}{2 x} \right )}}{3} - \frac{1}{x \sqrt{1 - \frac{9}{4 x^{2}}}} + \frac{9}{4 x^{3} \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**3,x)

[Out]

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

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Giac [A]  time = 2.56137, size = 39, normalized size = 1. \begin{align*} -\frac{\sqrt{4 \, x^{2} - 9}}{2 \, x^{2}} + \frac{2}{3} \, \arctan \left (\frac{1}{3} \, \sqrt{4 \, x^{2} - 9}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2-9)^(1/2)/x^3,x, algorithm="giac")

[Out]

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