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

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

[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

________________________________________________________________________________________

Rubi [A]  time = 0.0216427, 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, 207} \[ -\frac{\sqrt{4 x^2+9}}{18 x^2}-\frac{\sqrt{4 x^2+9}}{4 x^4}+\frac{2}{27} \tanh ^{-1}\left (\frac{1}{3} \sqrt{4 x^2+9}\right ) \]

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 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[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}{x^2 \sqrt{9+4 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}{9} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{9+4 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 = 0.0051901, size = 32, normalized size = 0.56 \[ -\frac{16 \left (4 x^2+9\right )^{3/2} \, _2F_1\left (\frac{3}{2},3;\frac{5}{2};\frac{4 x^2}{9}+1\right )}{2187} \]

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

________________________________________________________________________________________

Maple [A]  time = 0.006, 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))

________________________________________________________________________________________

Maxima [A]  time = 3.95155, size = 66, normalized size = 1.16 \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} \, \operatorname{arsinh}\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^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*arcsinh(3/2/abs(x))

________________________________________________________________________________________

Fricas [A]  time = 1.51197, size = 166, normalized size = 2.91 \begin{align*} \frac{8 \, x^{4} \log \left (-2 \, x + \sqrt{4 \, x^{2} + 9} + 3\right ) - 8 \, x^{4} \log \left (-2 \, x + \sqrt{4 \, x^{2} + 9} - 3\right ) - 3 \, \sqrt{4 \, x^{2} + 9}{\left (2 \, x^{2} + 9\right )}}{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(-2*x + sqrt(4*x^2 + 9) + 3) - 8*x^4*log(-2*x + sqrt(4*x^2 + 9) - 3) - 3*sqrt(4*x^2 + 9)*(2*x^
2 + 9))/x^4

________________________________________________________________________________________

Sympy [A]  time = 3.47304, size = 63, normalized size = 1.11 \begin{align*} \frac{2 \operatorname{asinh}{\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}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*asinh(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)))

________________________________________________________________________________________

Giac [A]  time = 2.68474, size = 74, normalized size = 1.3 \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)