3.198 \(\int \frac{2+3 x^2}{x^3 \left (3+5 x^2+x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=90 \[ -\frac{8 x^2+7}{39 x^2 \sqrt{x^4+5 x^2+3}}-\frac{2 \sqrt{x^4+5 x^2+3}}{39 x^2}+\frac{\tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right )}{3 \sqrt{3}} \]

[Out]

-(7 + 8*x^2)/(39*x^2*Sqrt[3 + 5*x^2 + x^4]) - (2*Sqrt[3 + 5*x^2 + x^4])/(39*x^2)
+ ArcTanh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4])]/(3*Sqrt[3])

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Rubi [A]  time = 0.20158, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{8 x^2+7}{39 x^2 \sqrt{x^4+5 x^2+3}}-\frac{2 \sqrt{x^4+5 x^2+3}}{39 x^2}+\frac{\tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(2 + 3*x^2)/(x^3*(3 + 5*x^2 + x^4)^(3/2)),x]

[Out]

-(7 + 8*x^2)/(39*x^2*Sqrt[3 + 5*x^2 + x^4]) - (2*Sqrt[3 + 5*x^2 + x^4])/(39*x^2)
+ ArcTanh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4])]/(3*Sqrt[3])

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Rubi in Sympy [A]  time = 20.9106, size = 80, normalized size = 0.89 \[ \frac{\sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (5 x^{2} + 6\right )}{6 \sqrt{x^{4} + 5 x^{2} + 3}} \right )}}{9} - \frac{8 x^{2} + 7}{39 x^{2} \sqrt{x^{4} + 5 x^{2} + 3}} - \frac{2 \sqrt{x^{4} + 5 x^{2} + 3}}{39 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3*x**2+2)/x**3/(x**4+5*x**2+3)**(3/2),x)

[Out]

sqrt(3)*atanh(sqrt(3)*(5*x**2 + 6)/(6*sqrt(x**4 + 5*x**2 + 3)))/9 - (8*x**2 + 7)
/(39*x**2*sqrt(x**4 + 5*x**2 + 3)) - 2*sqrt(x**4 + 5*x**2 + 3)/(39*x**2)

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Mathematica [A]  time = 0.0905552, size = 79, normalized size = 0.88 \[ \frac{\log \left (5 x^2+2 \sqrt{3} \sqrt{x^4+5 x^2+3}+6\right )-\log \left (x^2\right )}{3 \sqrt{3}}-\frac{2 x^4+18 x^2+13}{39 x^2 \sqrt{x^4+5 x^2+3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + 3*x^2)/(x^3*(3 + 5*x^2 + x^4)^(3/2)),x]

[Out]

-(13 + 18*x^2 + 2*x^4)/(39*x^2*Sqrt[3 + 5*x^2 + x^4]) + (-Log[x^2] + Log[6 + 5*x
^2 + 2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4]])/(3*Sqrt[3])

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Maple [A]  time = 0.023, size = 84, normalized size = 0.9 \[ -{\frac{1}{3\,{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}}}}-{\frac{1}{3}{\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}}}}-{\frac{2\,{x}^{2}+5}{39}{\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}}}}+{\frac{\sqrt{3}}{9}{\it Artanh} \left ({\frac{ \left ( 5\,{x}^{2}+6 \right ) \sqrt{3}}{6}{\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3*x^2+2)/x^3/(x^4+5*x^2+3)^(3/2),x)

[Out]

-1/3/x^2/(x^4+5*x^2+3)^(1/2)-1/3/(x^4+5*x^2+3)^(1/2)-1/39*(2*x^2+5)/(x^4+5*x^2+3
)^(1/2)+1/9*arctanh(1/6*(5*x^2+6)*3^(1/2)/(x^4+5*x^2+3)^(1/2))*3^(1/2)

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Maxima [A]  time = 0.780967, size = 111, normalized size = 1.23 \[ -\frac{2 \, x^{2}}{39 \, \sqrt{x^{4} + 5 \, x^{2} + 3}} + \frac{1}{9} \, \sqrt{3} \log \left (\frac{2 \, \sqrt{3} \sqrt{x^{4} + 5 \, x^{2} + 3}}{x^{2}} + \frac{6}{x^{2}} + 5\right ) - \frac{6}{13 \, \sqrt{x^{4} + 5 \, x^{2} + 3}} - \frac{1}{3 \, \sqrt{x^{4} + 5 \, x^{2} + 3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x^2 + 2)/((x^4 + 5*x^2 + 3)^(3/2)*x^3),x, algorithm="maxima")

[Out]

-2/39*x^2/sqrt(x^4 + 5*x^2 + 3) + 1/9*sqrt(3)*log(2*sqrt(3)*sqrt(x^4 + 5*x^2 + 3
)/x^2 + 6/x^2 + 5) - 6/13/sqrt(x^4 + 5*x^2 + 3) - 1/3/(sqrt(x^4 + 5*x^2 + 3)*x^2
)

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Fricas [A]  time = 0.282852, size = 333, normalized size = 3.7 \[ \frac{2 \, \sqrt{3}{\left (4 \, x^{4} + 15 \, x^{2} + 10\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} -{\left (8 \, x^{8} + 60 \, x^{6} + 124 \, x^{4} + 60 \, x^{2} -{\left (8 \, x^{6} + 40 \, x^{4} + 37 \, x^{2}\right )} \sqrt{x^{4} + 5 \, x^{2} + 3}\right )} \log \left (-\frac{6 \, x^{2} - \sqrt{3}{\left (2 \, x^{4} + 5 \, x^{2} + 6\right )} + 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (\sqrt{3} x^{2} - 3\right )}}{2 \, x^{4} - 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} x^{2} + 5 \, x^{2}}\right ) - \sqrt{3}{\left (8 \, x^{6} + 50 \, x^{4} + 82 \, x^{2} + 37\right )}}{3 \,{\left (\sqrt{3}{\left (8 \, x^{6} + 40 \, x^{4} + 37 \, x^{2}\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} - 4 \, \sqrt{3}{\left (2 \, x^{8} + 15 \, x^{6} + 31 \, x^{4} + 15 \, x^{2}\right )}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x^2 + 2)/((x^4 + 5*x^2 + 3)^(3/2)*x^3),x, algorithm="fricas")

[Out]

1/3*(2*sqrt(3)*(4*x^4 + 15*x^2 + 10)*sqrt(x^4 + 5*x^2 + 3) - (8*x^8 + 60*x^6 + 1
24*x^4 + 60*x^2 - (8*x^6 + 40*x^4 + 37*x^2)*sqrt(x^4 + 5*x^2 + 3))*log(-(6*x^2 -
sqrt(3)*(2*x^4 + 5*x^2 + 6) + 2*sqrt(x^4 + 5*x^2 + 3)*(sqrt(3)*x^2 - 3))/(2*x^4
- 2*sqrt(x^4 + 5*x^2 + 3)*x^2 + 5*x^2)) - sqrt(3)*(8*x^6 + 50*x^4 + 82*x^2 + 37
))/(sqrt(3)*(8*x^6 + 40*x^4 + 37*x^2)*sqrt(x^4 + 5*x^2 + 3) - 4*sqrt(3)*(2*x^8 +
15*x^6 + 31*x^4 + 15*x^2))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x**2+2)/x**3/(x**4+5*x**2+3)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{3 \, x^{2} + 2}{{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x^2 + 2)/((x^4 + 5*x^2 + 3)^(3/2)*x^3),x, algorithm="giac")

[Out]

integrate((3*x^2 + 2)/((x^4 + 5*x^2 + 3)^(3/2)*x^3), x)