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

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

[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 = 0.0657639, 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 \[ -\frac{\sqrt{4 x^2+9}}{18 x^2}+\frac{2}{27} \tanh ^{-1}\left (\frac{1}{3} \sqrt{4 x^2+9}\right )-\frac{\sqrt{4 x^2+9}}{4 x^4} \]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 6.68454, size = 46, normalized size = 0.81 \[ \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{4 x^{2} + 9}}{3} \right )}}{27} - \frac{\sqrt{4 x^{2} + 9}}{18 x^{2}} - \frac{\sqrt{4 x^{2} + 9}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((4*x**2+9)**(1/2)/x**5,x)

[Out]

2*atanh(sqrt(4*x**2 + 9)/3)/27 - sqrt(4*x**2 + 9)/(18*x**2) - sqrt(4*x**2 + 9)/(
4*x**4)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0408519, size = 48, normalized size = 0.84 \[ \frac{1}{108} \left (8 \log \left (\sqrt{4 x^2+9}+3\right )-\frac{3 \sqrt{4 x^2+9} \left (2 x^2+9\right )}{x^4}-8 \log (x)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[9 + 4*x^2]/x^5,x]

[Out]

((-3*(9 + 2*x^2)*Sqrt[9 + 4*x^2])/x^4 - 8*Log[x] + 8*Log[3 + Sqrt[9 + 4*x^2]])/1
08

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 55, normalized size = 1. \[ -{\frac{1}{36\,{x}^{4}} \left ( 4\,{x}^{2}+9 \right ) ^{{\frac{3}{2}}}}+{\frac{1}{162\,{x}^{2}} \left ( 4\,{x}^{2}+9 \right ) ^{{\frac{3}{2}}}}-{\frac{2}{81}\sqrt{4\,{x}^{2}+9}}+{\frac{2}{27}{\it Artanh} \left ( 3\,{\frac{1}{\sqrt{4\,{x}^{2}+9}}} \right ) } \]

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*ar
ctanh(3/(4*x^2+9)^(1/2))

_______________________________________________________________________________________

Maxima [A]  time = 1.49798, size = 66, normalized size = 1.16 \[ -\frac{2}{81} \, \sqrt{4 \, x^{2} + 9} + \frac{{\left (4 \, x^{2} + 9\right )}^{\frac{3}{2}}}{162 \, x^{2}} - \frac{{\left (4 \, x^{2} + 9\right )}^{\frac{3}{2}}}{36 \, x^{4}} + \frac{2}{27} \, \operatorname{arsinh}\left (\frac{3}{2 \,{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(4*x^2 + 9)/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 = 0.22987, size = 269, normalized size = 4.72 \[ \frac{1536 \, x^{7} + 12096 \, x^{5} + 27216 \, x^{3} + 8 \,{\left (128 \, x^{8} + 288 \, x^{6} + 81 \, x^{4} - 8 \,{\left (8 \, x^{7} + 9 \, x^{5}\right )} \sqrt{4 \, x^{2} + 9}\right )} \log \left (-2 \, x + \sqrt{4 \, x^{2} + 9} + 3\right ) - 8 \,{\left (128 \, x^{8} + 288 \, x^{6} + 81 \, x^{4} - 8 \,{\left (8 \, x^{7} + 9 \, x^{5}\right )} \sqrt{4 \, x^{2} + 9}\right )} \log \left (-2 \, x + \sqrt{4 \, x^{2} + 9} - 3\right ) - 3 \,{\left (256 \, x^{6} + 1728 \, x^{4} + 2754 \, x^{2} + 729\right )} \sqrt{4 \, x^{2} + 9} + 17496 \, x}{108 \,{\left (128 \, x^{8} + 288 \, x^{6} + 81 \, x^{4} - 8 \,{\left (8 \, x^{7} + 9 \, x^{5}\right )} \sqrt{4 \, x^{2} + 9}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(4*x^2 + 9)/x^5,x, algorithm="fricas")

[Out]

1/108*(1536*x^7 + 12096*x^5 + 27216*x^3 + 8*(128*x^8 + 288*x^6 + 81*x^4 - 8*(8*x
^7 + 9*x^5)*sqrt(4*x^2 + 9))*log(-2*x + sqrt(4*x^2 + 9) + 3) - 8*(128*x^8 + 288*
x^6 + 81*x^4 - 8*(8*x^7 + 9*x^5)*sqrt(4*x^2 + 9))*log(-2*x + sqrt(4*x^2 + 9) - 3
) - 3*(256*x^6 + 1728*x^4 + 2754*x^2 + 729)*sqrt(4*x^2 + 9) + 17496*x)/(128*x^8
+ 288*x^6 + 81*x^4 - 8*(8*x^7 + 9*x^5)*sqrt(4*x^2 + 9))

_______________________________________________________________________________________

Sympy [A]  time = 12.4241, size = 63, normalized size = 1.11 \[ \frac{2 \operatorname{asinh}{\left (\frac{3}{2 x} \right )}}{27} - \frac{1}{9 x \sqrt{1 + \frac{9}{4 x^{2}}}} - \frac{3}{4 x^{3} \sqrt{1 + \frac{9}{4 x^{2}}}} - \frac{9}{8 x^{5} \sqrt{1 + \frac{9}{4 x^{2}}}} \]

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/XCAS [A]  time = 0.205551, size = 74, normalized size = 1.3 \[ -\frac{{\left (4 \, x^{2} + 9\right )}^{\frac{3}{2}} + 9 \, \sqrt{4 \, x^{2} + 9}}{72 \, x^{4}} + \frac{1}{27} \,{\rm ln}\left (\sqrt{4 \, x^{2} + 9} + 3\right ) - \frac{1}{27} \,{\rm ln}\left (\sqrt{4 \, x^{2} + 9} - 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(4*x^2 + 9)/x^5,x, algorithm="giac")

[Out]

-1/72*((4*x^2 + 9)^(3/2) + 9*sqrt(4*x^2 + 9))/x^4 + 1/27*ln(sqrt(4*x^2 + 9) + 3)
- 1/27*ln(sqrt(4*x^2 + 9) - 3)