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

Optimal. Leaf size=30 \[ \sqrt{-4 x^2-9}-3 \tan ^{-1}\left (\frac{1}{3} \sqrt{-4 x^2-9}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0151617, antiderivative size = 30, 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, 50, 63, 204} \[ \sqrt{-4 x^2-9}-3 \tan ^{-1}\left (\frac{1}{3} \sqrt{-4 x^2-9}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[-9 - 4*x^2]/x,x]

[Out]

Sqrt[-9 - 4*x^2] - 3*ArcTan[Sqrt[-9 - 4*x^2]/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 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 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 204

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

Rubi steps

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

Mathematica [A]  time = 0.0043035, size = 30, normalized size = 1. \[ \sqrt{-4 x^2-9}-3 \tan ^{-1}\left (\frac{1}{3} \sqrt{-4 x^2-9}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-9 - 4*x^2]/x,x]

[Out]

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

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Maple [A]  time = 0.004, size = 25, normalized size = 0.8 \begin{align*} \sqrt{-4\,{x}^{2}-9}+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(1/x*(-4*x^2-9)^(1/2),x)

[Out]

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

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Maxima [C]  time = 3.30422, size = 47, normalized size = 1.57 \begin{align*} \sqrt{-4 \, x^{2} - 9} + 3 i \, \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(1/x*(-4*x^2-9)^(1/2),x, algorithm="maxima")

[Out]

sqrt(-4*x^2 - 9) + 3*I*log(6*sqrt(4*x^2 + 9)/abs(x) + 18/abs(x))

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Fricas [C]  time = 1.289, size = 142, normalized size = 4.73 \begin{align*} \sqrt{-4 \, x^{2} - 9} - \frac{3}{2} i \, \log \left (-\frac{6 \,{\left (i \, \sqrt{-4 \, x^{2} - 9} - 3\right )}}{x}\right ) + \frac{3}{2} i \, \log \left (-\frac{6 \,{\left (-i \, \sqrt{-4 \, x^{2} - 9} - 3\right )}}{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(-4*x^2 - 9) - 3/2*I*log(-6*(I*sqrt(-4*x^2 - 9) - 3)/x) + 3/2*I*log(-6*(-I*sqrt(-4*x^2 - 9) - 3)/x)

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Sympy [C]  time = 1.24054, size = 44, normalized size = 1.47 \begin{align*} \frac{2 i x}{\sqrt{1 + \frac{9}{4 x^{2}}}} - 3 i \operatorname{asinh}{\left (\frac{3}{2 x} \right )} + \frac{9 i}{2 x \sqrt{1 + \frac{9}{4 x^{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*I*x/sqrt(1 + 9/(4*x**2)) - 3*I*asinh(3/(2*x)) + 9*I/(2*x*sqrt(1 + 9/(4*x**2)))

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Giac [C]  time = 2.42821, size = 35, normalized size = 1.17 \begin{align*} i \, \sqrt{4 \, x^{2} + 9} - 3 \, \arctan \left (\frac{1}{3} i \, \sqrt{4 \, x^{2} + 9}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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