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

Optimal. Leaf size=66 \[ -\frac{8 x^2+7}{39 \sqrt{x^4+5 x^2+3}}-\frac{\tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right )}{3 \sqrt{3}} \]

[Out]

-(7 + 8*x^2)/(39*Sqrt[3 + 5*x^2 + x^4]) - 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 = 0.148657, antiderivative size = 66, 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 \[ -\frac{8 x^2+7}{39 \sqrt{x^4+5 x^2+3}}-\frac{\tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

-(7 + 8*x^2)/(39*Sqrt[3 + 5*x^2 + x^4]) - 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 = 17.7022, size = 58, normalized size = 0.88 \[ - \frac{8 x^{2} + 7}{39 \sqrt{x^{4} + 5 x^{2} + 3}} - \frac{\sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (5 x^{2} + 6\right )}{6 \sqrt{x^{4} + 5 x^{2} + 3}} \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.127906, size = 71, normalized size = 1.08 \[ \frac{-8 x^2-7}{39 \sqrt{x^4+5 x^2+3}}+\frac{\log \left (x^2\right )-\log \left (5 x^2+2 \sqrt{3} \sqrt{x^4+5 x^2+3}+6\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(-7 - 8*x^2)/(39*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 = 0.027, size = 67, normalized size = 1. \[{\frac{1}{3}{\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}}}}-{\frac{8\,{x}^{2}+20}{39}{\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}}}}-{\frac{\sqrt{3}}{9}{\it Artanh} \left ({\frac{ \left ( 5\,{x}^{2}+6 \right ) \sqrt{3}}{6}{\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3/(x^4+5*x^2+3)^(1/2)-4/39*(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 = 0.773884, size = 88, normalized size = 1.33 \[ -\frac{8 \, x^{2}}{39 \, \sqrt{x^{4} + 5 \, x^{2} + 3}} - \frac{1}{9} \, \sqrt{3} \log \left (\frac{2 \, \sqrt{3} \sqrt{x^{4} + 5 \, x^{2} + 3}}{x^{2}} + \frac{6}{x^{2}} + 5\right ) - \frac{7}{39 \, \sqrt{x^{4} + 5 \, x^{2} + 3}} \]

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),x, algorithm="maxima")

[Out]

-8/39*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) - 7/39/sqrt(x^4 + 5*x^2 + 3)

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Fricas [A]  time = 0.268539, size = 248, normalized size = 3.76 \[ -\frac{{\left (2 \, x^{4} + 10 \, x^{2} - \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (2 \, x^{2} + 5\right )} + 6\right )} \log \left (\frac{6 \, x^{2} + \sqrt{3}{\left (2 \, x^{4} + 5 \, x^{2} + 6\right )} - 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (\sqrt{3} x^{2} + 3\right )}}{2 \, x^{4} - 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} x^{2} + 5 \, x^{2}}\right ) - \sqrt{3}{\left (2 \, x^{2} + 1\right )} + 2 \, \sqrt{3} \sqrt{x^{4} + 5 \, x^{2} + 3}}{3 \,{\left (\sqrt{3} \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (2 \, x^{2} + 5\right )} - 2 \, \sqrt{3}{\left (x^{4} + 5 \, x^{2} + 3\right )}\right )}} \]

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),x, algorithm="fricas")

[Out]

-1/3*((2*x^4 + 10*x^2 - sqrt(x^4 + 5*x^2 + 3)*(2*x^2 + 5) + 6)*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)*(2*x^2 + 1) + 2*sqrt(3)*sqrt(x^4 +
5*x^2 + 3))/(sqrt(3)*sqrt(x^4 + 5*x^2 + 3)*(2*x^2 + 5) - 2*sqrt(3)*(x^4 + 5*x^2
+ 3))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((3*x**2 + 2)/(x*(x**4 + 5*x**2 + 3)**(3/2)), x)

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

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),x, algorithm="giac")

[Out]

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