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

Optimal. Leaf size=69 \[ \frac{3}{2} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right )-\frac{\tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right )}{\sqrt{3}} \]

[Out]

(3*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5*x^2 + x^4])])/2 - ArcTanh[(6 + 5*x^2)/(2*Sq
rt[3]*Sqrt[3 + 5*x^2 + x^4])]/Sqrt[3]

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Rubi [A]  time = 0.145927, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{3}{2} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right )-\frac{\tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(3*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5*x^2 + x^4])])/2 - ArcTanh[(6 + 5*x^2)/(2*Sq
rt[3]*Sqrt[3 + 5*x^2 + x^4])]/Sqrt[3]

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Rubi in Sympy [A]  time = 16.8047, size = 61, normalized size = 0.88 \[ \frac{3 \operatorname{atanh}{\left (\frac{2 x^{2} + 5}{2 \sqrt{x^{4} + 5 x^{2} + 3}} \right )}}{2} - \frac{\sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (5 x^{2} + 6\right )}{6 \sqrt{x^{4} + 5 x^{2} + 3}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.112107, size = 78, normalized size = 1.13 \[ \frac{3}{2} \log \left (2 x^2+2 \sqrt{x^4+5 x^2+3}+5\right )+2 \left (\frac{\log (x)}{\sqrt{3}}-\frac{\log \left (5 x^2+2 \sqrt{3} \sqrt{x^4+5 x^2+3}+6\right )}{2 \sqrt{3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.016, size = 52, normalized size = 0.8 \[ -{\frac{\sqrt{3}}{3}{\it Artanh} \left ({\frac{ \left ( 5\,{x}^{2}+6 \right ) \sqrt{3}}{6}{\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}}}} \right ) }+{\frac{3}{2}\ln \left ({x}^{2}+{\frac{5}{2}}+\sqrt{{x}^{4}+5\,{x}^{2}+3} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.780192, size = 78, normalized size = 1.13 \[ -\frac{1}{3} \, \sqrt{3} \log \left (\frac{2 \, \sqrt{3} \sqrt{x^{4} + 5 \, x^{2} + 3}}{x^{2}} + \frac{6}{x^{2}} + 5\right ) + \frac{3}{2} \, \log \left (2 \, x^{2} + 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} + 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.280924, size = 151, normalized size = 2.19 \[ -\frac{1}{6} \, \sqrt{3}{\left (3 \, \sqrt{3} \log \left (-2 \, x^{2} + 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} - 5\right ) - 2 \, \log \left (\frac{6 \, x^{2} + \sqrt{3}{\left (2 \, x^{4} + 5 \, x^{2} + 6\right )} - 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (\sqrt{3} x^{2} + 3\right )}}{2 \, x^{4} - 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} x^{2} + 5 \, x^{2}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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