3.542 \(\int \frac{1}{x^5 \sqrt{9+4 x^2}} \, dx\)

Optimal. Leaf size=57 \[ \frac{\sqrt{4 x^2+9}}{54 x^2}-\frac{2}{81} \tanh ^{-1}\left (\frac{1}{3} \sqrt{4 x^2+9}\right )-\frac{\sqrt{4 x^2+9}}{36 x^4} \]

[Out]

-Sqrt[9 + 4*x^2]/(36*x^4) + Sqrt[9 + 4*x^2]/(54*x^2) - (2*ArcTanh[Sqrt[9 + 4*x^2
]/3])/81

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Rubi [A]  time = 0.0699204, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{\sqrt{4 x^2+9}}{54 x^2}-\frac{2}{81} \tanh ^{-1}\left (\frac{1}{3} \sqrt{4 x^2+9}\right )-\frac{\sqrt{4 x^2+9}}{36 x^4} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*Sqrt[9 + 4*x^2]),x]

[Out]

-Sqrt[9 + 4*x^2]/(36*x^4) + Sqrt[9 + 4*x^2]/(54*x^2) - (2*ArcTanh[Sqrt[9 + 4*x^2
]/3])/81

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Rubi in Sympy [A]  time = 6.56026, size = 46, normalized size = 0.81 \[ - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{4 x^{2} + 9}}{3} \right )}}{81} + \frac{\sqrt{4 x^{2} + 9}}{54 x^{2}} - \frac{\sqrt{4 x^{2} + 9}}{36 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*atanh(sqrt(4*x**2 + 9)/3)/81 + sqrt(4*x**2 + 9)/(54*x**2) - sqrt(4*x**2 + 9)/
(36*x**4)

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Mathematica [A]  time = 0.0395131, size = 48, normalized size = 0.84 \[ \frac{1}{324} \left (-8 \log \left (\sqrt{4 x^2+9}+3\right )+\frac{3 \sqrt{4 x^2+9} \left (2 x^2-3\right )}{x^4}+8 \log (x)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^5*Sqrt[9 + 4*x^2]),x]

[Out]

((3*(-3 + 2*x^2)*Sqrt[9 + 4*x^2])/x^4 + 8*Log[x] - 8*Log[3 + Sqrt[9 + 4*x^2]])/3
24

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Maple [A]  time = 0.007, size = 44, normalized size = 0.8 \[ -{\frac{1}{36\,{x}^{4}}\sqrt{4\,{x}^{2}+9}}+{\frac{1}{54\,{x}^{2}}\sqrt{4\,{x}^{2}+9}}-{\frac{2}{81}{\it Artanh} \left ( 3\,{\frac{1}{\sqrt{4\,{x}^{2}+9}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(4*x^2+9)^(1/2),x)

[Out]

-1/36*(4*x^2+9)^(1/2)/x^4+1/54*(4*x^2+9)^(1/2)/x^2-2/81*arctanh(3/(4*x^2+9)^(1/2
))

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Maxima [A]  time = 1.50026, size = 51, normalized size = 0.89 \[ \frac{\sqrt{4 \, x^{2} + 9}}{54 \, x^{2}} - \frac{\sqrt{4 \, x^{2} + 9}}{36 \, x^{4}} - \frac{2}{81} \, \operatorname{arsinh}\left (\frac{3}{2 \,{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(4*x^2 + 9)*x^5),x, algorithm="maxima")

[Out]

1/54*sqrt(4*x^2 + 9)/x^2 - 1/36*sqrt(4*x^2 + 9)/x^4 - 2/81*arcsinh(3/2/abs(x))

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Fricas [A]  time = 0.23085, size = 269, normalized size = 4.72 \[ -\frac{1536 \, x^{7} + 2880 \, x^{5} - 3888 \, x^{3} + 8 \,{\left (128 \, x^{8} + 288 \, x^{6} + 81 \, x^{4} - 8 \,{\left (8 \, x^{7} + 9 \, x^{5}\right )} \sqrt{4 \, x^{2} + 9}\right )} \log \left (-2 \, x + \sqrt{4 \, x^{2} + 9} + 3\right ) - 8 \,{\left (128 \, x^{8} + 288 \, x^{6} + 81 \, x^{4} - 8 \,{\left (8 \, x^{7} + 9 \, x^{5}\right )} \sqrt{4 \, x^{2} + 9}\right )} \log \left (-2 \, x + \sqrt{4 \, x^{2} + 9} - 3\right ) - 3 \,{\left (256 \, x^{6} + 192 \, x^{4} - 702 \, x^{2} - 243\right )} \sqrt{4 \, x^{2} + 9} - 5832 \, x}{324 \,{\left (128 \, x^{8} + 288 \, x^{6} + 81 \, x^{4} - 8 \,{\left (8 \, x^{7} + 9 \, x^{5}\right )} \sqrt{4 \, x^{2} + 9}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(4*x^2 + 9)*x^5),x, algorithm="fricas")

[Out]

-1/324*(1536*x^7 + 2880*x^5 - 3888*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 + 192*x^4 - 702*x^2 - 243)*sqrt(4*x^2 + 9) - 5832*x)/(128*x^8 + 28
8*x^6 + 81*x^4 - 8*(8*x^7 + 9*x^5)*sqrt(4*x^2 + 9))

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Sympy [A]  time = 16.8939, size = 63, normalized size = 1.11 \[ - \frac{2 \operatorname{asinh}{\left (\frac{3}{2 x} \right )}}{81} + \frac{1}{27 x \sqrt{1 + \frac{9}{4 x^{2}}}} + \frac{1}{36 x^{3} \sqrt{1 + \frac{9}{4 x^{2}}}} - \frac{1}{8 x^{5} \sqrt{1 + \frac{9}{4 x^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*asinh(3/(2*x))/81 + 1/(27*x*sqrt(1 + 9/(4*x**2))) + 1/(36*x**3*sqrt(1 + 9/(4*
x**2))) - 1/(8*x**5*sqrt(1 + 9/(4*x**2)))

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GIAC/XCAS [A]  time = 0.215255, size = 74, normalized size = 1.3 \[ \frac{{\left (4 \, x^{2} + 9\right )}^{\frac{3}{2}} - 15 \, \sqrt{4 \, x^{2} + 9}}{216 \, x^{4}} - \frac{1}{81} \,{\rm ln}\left (\sqrt{4 \, x^{2} + 9} + 3\right ) + \frac{1}{81} \,{\rm ln}\left (\sqrt{4 \, x^{2} + 9} - 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(4*x^2 + 9)*x^5),x, algorithm="giac")

[Out]

1/216*((4*x^2 + 9)^(3/2) - 15*sqrt(4*x^2 + 9))/x^4 - 1/81*ln(sqrt(4*x^2 + 9) + 3
) + 1/81*ln(sqrt(4*x^2 + 9) - 3)