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

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 = 0.0165918, 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, 203} \[ \sqrt{4 x^2-9}-3 \tan ^{-1}\left (\frac{1}{3} \sqrt{4 x^2-9}\right ) \]

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 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} \, 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}{x \sqrt{-9+4 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 = 0.0042368, 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((4*x^2-9)^(1/2)/x,x)

[Out]

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

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Maxima [A]  time = 4.40072, size = 26, normalized size = 0.87 \begin{align*} \sqrt{4 \, x^{2} - 9} + 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,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.49895, size = 78, normalized size = 2.6 \begin{align*} \sqrt{4 \, x^{2} - 9} - 6 \, \arctan \left (-\frac{2}{3} \, x + \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,x, algorithm="fricas")

[Out]

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

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Sympy [C]  time = 1.33413, size = 82, normalized size = 2.73 \begin{align*} \begin{cases} \sqrt{4 x^{2} - 9} - 3 i \log{\left (x \right )} + \frac{3 i \log{\left (x^{2} \right )}}{2} + 3 \operatorname{asin}{\left (\frac{3}{2 x} \right )} & \text{for}\: \frac{4 \left |{x^{2}}\right |}{9} > 1 \\i \sqrt{9 - 4 x^{2}} + \frac{3 i \log{\left (x^{2} \right )}}{2} - 3 i \log{\left (\sqrt{1 - \frac{4 x^{2}}{9}} + 1 \right )} & \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,x)

[Out]

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

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Giac [A]  time = 2.69534, size = 32, normalized size = 1.07 \begin{align*} \sqrt{4 \, x^{2} - 9} - 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,x, algorithm="giac")

[Out]

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