3.5.82 \(\int \frac {\sqrt {-9-4 x^2}}{x^3} \, dx\) [482]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 39, normalized size of antiderivative = 1.00, 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 = {272, 43, 65, 210} \begin {gather*} -\frac {2}{3} \text {ArcTan}\left (\frac {1}{3} \sqrt {-4 x^2-9}\right )-\frac {\sqrt {-4 x^2-9}}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[-9 - 4*x^2]/x^3,x]

[Out]

-1/2*Sqrt[-9 - 4*x^2]/x^2 - (2*ArcTan[Sqrt[-9 - 4*x^2]/3])/3

Rule 43

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, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

Rule 65

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 210

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

Rule 272

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]]

Rubi steps

\begin {align*} \int \frac {\sqrt {-9-4 x^2}}{x^3} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {-9-4 x}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {-9-4 x^2}}{2 x^2}-\text {Subst}\left (\int \frac {1}{\sqrt {-9-4 x} x} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {-9-4 x^2}}{2 x^2}+\frac {1}{2} \text {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 = 0.02, size = 39, normalized size = 1.00 \begin {gather*} -\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 {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-9 - 4*x^2]/x^3,x]

[Out]

-1/2*Sqrt[-9 - 4*x^2]/x^2 - (2*ArcTan[Sqrt[-9 - 4*x^2]/3])/3

________________________________________________________________________________________

Maple [A]
time = 0.14, size = 41, normalized size = 1.05

method	result	size
risch	\(\frac {4 x^{2}+9}{2 x^{2} \sqrt {-4 x^{2}-9}}+\frac {2 \arctan \left (\frac {3}{\sqrt {-4 x^{2}-9}}\right )}{3}\)	\(37\)
default	\(\frac {\left (-4 x^{2}-9\right )^{\frac {3}{2}}}{18 x^{2}}+\frac {2 \sqrt {-4 x^{2}-9}}{9}+\frac {2 \arctan \left (\frac {3}{\sqrt {-4 x^{2}-9}}\right )}{3}\)	\(41\)
trager	\(-\frac {\sqrt {-4 x^{2}-9}}{2 x^{2}}-\frac {2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\sqrt {-4 x^{2}-9}+3 \RootOf \left (\textit {\_Z}^{2}+1\right )}{x}\right )}{3}\)	\(47\)
meijerg	\(-\frac {i \left (-\frac {9 \sqrt {\pi }\, \left (8+\frac {16 x^{2}}{9}\right )}{16 x^{2}}+\frac {9 \sqrt {\pi }\, \sqrt {1+\frac {4 x^{2}}{9}}}{2 x^{2}}+2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {1+\frac {4 x^{2}}{9}}}{2}\right )-\left (-1+2 \ln \left (x \right )-2 \ln \left (3\right )\right ) \sqrt {\pi }+\frac {9 \sqrt {\pi }}{2 x^{2}}\right )}{3 \sqrt {\pi }}\)	\(82\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*x^2-9)^(1/2)/x^3,x,method=_RETURNVERBOSE)

[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))

________________________________________________________________________________________

Maxima [C] Result contains complex when optimal does not.
time = 0.58, size = 51, normalized size = 1.31 \begin {gather*} \frac {2}{9} \, \sqrt {-4 \, x^{2} - 9} + \frac {{\left (-4 \, x^{2} - 9\right )}^{\frac {3}{2}}}{18 \, x^{2}} + \frac {2}{3} i \, \log \left (\frac {6 \, \sqrt {4 \, x^{2} + 9}}{{\left | x \right |}} + \frac {18}{{\left | x \right |}}\right ) \end {gather*}

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

________________________________________________________________________________________

Fricas [C] Result contains complex when optimal does not.
time = 0.76, size = 65, normalized size = 1.67 \begin {gather*} \frac {-2 i \, x^{2} \log \left (-\frac {4 \, {\left (i \, \sqrt {-4 \, x^{2} - 9} - 3\right )}}{3 \, x}\right ) + 2 i \, x^{2} \log \left (-\frac {4 \, {\left (-i \, \sqrt {-4 \, x^{2} - 9} - 3\right )}}{3 \, x}\right ) - 3 \, \sqrt {-4 \, x^{2} - 9}}{6 \, x^{2}} \end {gather*}

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

________________________________________________________________________________________

Sympy [C] Result contains complex when optimal does not.
time = 0.82, size = 27, normalized size = 0.69 \begin {gather*} - \frac {2 i \operatorname {asinh}{\left (\frac {3}{2 x} \right )}}{3} - \frac {i \sqrt {1 + \frac {9}{4 x^{2}}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x**2-9)**(1/2)/x**3,x)

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 0.53, size = 29, normalized size = 0.74 \begin {gather*} -\frac {\sqrt {-4 \, x^{2} - 9}}{2 \, x^{2}} - \frac {2}{3} \, \arctan \left (\frac {1}{3} \, \sqrt {-4 \, x^{2} - 9}\right ) \end {gather*}

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))

________________________________________________________________________________________

Mupad [B]
time = 4.79, size = 29, normalized size = 0.74 \begin {gather*} -\frac {2\,\mathrm {atan}\left (\frac {\sqrt {-4\,x^2-9}}{3}\right )}{3}-\frac {\sqrt {-4\,x^2-9}}{2\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((- 4*x^2 - 9)^(1/2)/x^3,x)

[Out]

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

________________________________________________________________________________________