3.187 \(\int \frac{2+3 x^2}{x^5 \sqrt{3+5 x^2+x^4}} \, dx\)

Optimal. Leaf size=83 \[ -\frac{\sqrt{x^4+5 x^2+3}}{12 x^2}-\frac{\sqrt{x^4+5 x^2+3}}{6 x^4}+\frac{1}{8} \sqrt{3} \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right ) \]

[Out]

-Sqrt[3 + 5*x^2 + x^4]/(6*x^4) - Sqrt[3 + 5*x^2 + x^4]/(12*x^2) + (Sqrt[3]*ArcTa
nh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4])])/8

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Rubi [A]  time = 0.186838, antiderivative size = 83, 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 \[ -\frac{\sqrt{x^4+5 x^2+3}}{12 x^2}-\frac{\sqrt{x^4+5 x^2+3}}{6 x^4}+\frac{1}{8} \sqrt{3} \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[3 + 5*x^2 + x^4]/(6*x^4) - Sqrt[3 + 5*x^2 + x^4]/(12*x^2) + (Sqrt[3]*ArcTa
nh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4])])/8

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Rubi in Sympy [A]  time = 20.3406, size = 71, normalized size = 0.86 \[ \frac{\sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (5 x^{2} + 6\right )}{6 \sqrt{x^{4} + 5 x^{2} + 3}} \right )}}{8} - \frac{\sqrt{x^{4} + 5 x^{2} + 3}}{12 x^{2}} - \frac{\sqrt{x^{4} + 5 x^{2} + 3}}{6 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3*x**2+2)/x**5/(x**4+5*x**2+3)**(1/2),x)

[Out]

sqrt(3)*atanh(sqrt(3)*(5*x**2 + 6)/(6*sqrt(x**4 + 5*x**2 + 3)))/8 - sqrt(x**4 +
5*x**2 + 3)/(12*x**2) - sqrt(x**4 + 5*x**2 + 3)/(6*x**4)

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Mathematica [A]  time = 0.0896247, size = 72, normalized size = 0.87 \[ \frac{1}{8} \sqrt{3} \left (\log \left (5 x^2+2 \sqrt{3} \sqrt{x^4+5 x^2+3}+6\right )-\log \left (x^2\right )\right )-\frac{\left (x^2+2\right ) \sqrt{x^4+5 x^2+3}}{12 x^4} \]

Antiderivative was successfully verified.

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

[Out]

-((2 + x^2)*Sqrt[3 + 5*x^2 + x^4])/(12*x^4) + (Sqrt[3]*(-Log[x^2] + Log[6 + 5*x^
2 + 2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4]]))/8

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Maple [A]  time = 0.019, size = 66, normalized size = 0.8 \[{\frac{\sqrt{3}}{8}{\it Artanh} \left ({\frac{ \left ( 5\,{x}^{2}+6 \right ) \sqrt{3}}{6}{\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}}}} \right ) }-{\frac{1}{6\,{x}^{4}}\sqrt{{x}^{4}+5\,{x}^{2}+3}}-{\frac{1}{12\,{x}^{2}}\sqrt{{x}^{4}+5\,{x}^{2}+3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3*x^2+2)/x^5/(x^4+5*x^2+3)^(1/2),x)

[Out]

1/8*arctanh(1/6*(5*x^2+6)*3^(1/2)/(x^4+5*x^2+3)^(1/2))*3^(1/2)-1/6*(x^4+5*x^2+3)
^(1/2)/x^4-1/12*(x^4+5*x^2+3)^(1/2)/x^2

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Maxima [A]  time = 0.780186, size = 92, normalized size = 1.11 \[ \frac{1}{8} \, \sqrt{3} \log \left (\frac{2 \, \sqrt{3} \sqrt{x^{4} + 5 \, x^{2} + 3}}{x^{2}} + \frac{6}{x^{2}} + 5\right ) - \frac{\sqrt{x^{4} + 5 \, x^{2} + 3}}{12 \, x^{2}} - \frac{\sqrt{x^{4} + 5 \, x^{2} + 3}}{6 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*sqrt(3)*log(2*sqrt(3)*sqrt(x^4 + 5*x^2 + 3)/x^2 + 6/x^2 + 5) - 1/12*sqrt(x^4
+ 5*x^2 + 3)/x^2 - 1/6*sqrt(x^4 + 5*x^2 + 3)/x^4

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Fricas [A]  time = 0.267834, size = 290, normalized size = 3.49 \[ \frac{72 \, x^{6} + 414 \, x^{4} + 616 \, x^{2} - 3 \,{\left (4 \, \sqrt{3}{\left (2 \, x^{6} + 5 \, x^{4}\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} - \sqrt{3}{\left (8 \, x^{8} + 40 \, x^{6} + 37 \, x^{4}\right )}\right )} \log \left (\frac{2 \, x^{4} - 2 \, \sqrt{3} x^{2} + 5 \, x^{2} - 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (x^{2} - \sqrt{3}\right )} + 6}{2 \, x^{4} - 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} x^{2} + 5 \, x^{2}}\right ) - 2 \,{\left (36 \, x^{4} + 117 \, x^{2} + 74\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} + 240}{24 \,{\left (8 \, x^{8} + 40 \, x^{6} + 37 \, x^{4} - 4 \,{\left (2 \, x^{6} + 5 \, x^{4}\right )} \sqrt{x^{4} + 5 \, x^{2} + 3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*(72*x^6 + 414*x^4 + 616*x^2 - 3*(4*sqrt(3)*(2*x^6 + 5*x^4)*sqrt(x^4 + 5*x^2
+ 3) - sqrt(3)*(8*x^8 + 40*x^6 + 37*x^4))*log((2*x^4 - 2*sqrt(3)*x^2 + 5*x^2 -
2*sqrt(x^4 + 5*x^2 + 3)*(x^2 - sqrt(3)) + 6)/(2*x^4 - 2*sqrt(x^4 + 5*x^2 + 3)*x^
2 + 5*x^2)) - 2*(36*x^4 + 117*x^2 + 74)*sqrt(x^4 + 5*x^2 + 3) + 240)/(8*x^8 + 40
*x^6 + 37*x^4 - 4*(2*x^6 + 5*x^4)*sqrt(x^4 + 5*x^2 + 3))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x**2+2)/x**5/(x**4+5*x**2+3)**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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