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

Optimal. Leaf size=97 \[ -\frac{\sqrt{x^4+5 x^2+3} \left (2-3 x^2\right )}{2 x^2}+\frac{19}{4} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right )-\frac{7 \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right )}{\sqrt{3}} \]

[Out]

-((2 - 3*x^2)*Sqrt[3 + 5*x^2 + x^4])/(2*x^2) + (19*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3
+ 5*x^2 + x^4])])/4 - (7*ArcTanh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4])]
)/Sqrt[3]

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Rubi [A]  time = 0.208238, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ -\frac{\sqrt{x^4+5 x^2+3} \left (2-3 x^2\right )}{2 x^2}+\frac{19}{4} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right )-\frac{7 \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

-((2 - 3*x^2)*Sqrt[3 + 5*x^2 + x^4])/(2*x^2) + (19*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3
+ 5*x^2 + x^4])])/4 - (7*ArcTanh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4])]
)/Sqrt[3]

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Rubi in Sympy [A]  time = 21.1162, size = 88, normalized size = 0.91 \[ \frac{19 \operatorname{atanh}{\left (\frac{2 x^{2} + 5}{2 \sqrt{x^{4} + 5 x^{2} + 3}} \right )}}{4} - \frac{7 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (5 x^{2} + 6\right )}{6 \sqrt{x^{4} + 5 x^{2} + 3}} \right )}}{3} - \frac{\left (- 3 x^{2} + 2\right ) \sqrt{x^{4} + 5 x^{2} + 3}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

19*atanh((2*x**2 + 5)/(2*sqrt(x**4 + 5*x**2 + 3)))/4 - 7*sqrt(3)*atanh(sqrt(3)*(
5*x**2 + 6)/(6*sqrt(x**4 + 5*x**2 + 3)))/3 - (-3*x**2 + 2)*sqrt(x**4 + 5*x**2 +
3)/(2*x**2)

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Mathematica [A]  time = 0.130576, size = 96, normalized size = 0.99 \[ \sqrt{x^4+5 x^2+3} \left (\frac{3}{2}-\frac{1}{x^2}\right )+\frac{19}{4} \log \left (2 x^2+2 \sqrt{x^4+5 x^2+3}+5\right )+\frac{7 \left (2 \log (x)-\log \left (5 x^2+2 \sqrt{3} \sqrt{x^4+5 x^2+3}+6\right )\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(3/2 - x^(-2))*Sqrt[3 + 5*x^2 + x^4] + (19*Log[5 + 2*x^2 + 2*Sqrt[3 + 5*x^2 + x^
4]])/4 + (7*(2*Log[x] - Log[6 + 5*x^2 + 2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4]]))/Sqrt[
3]

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Maple [A]  time = 0.022, size = 104, normalized size = 1.1 \[ -{\frac{1}{3\,{x}^{2}} \left ({x}^{4}+5\,{x}^{2}+3 \right ) ^{{\frac{3}{2}}}}+{\frac{7}{3}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{19}{4}\ln \left ({x}^{2}+{\frac{5}{2}}+\sqrt{{x}^{4}+5\,{x}^{2}+3} \right ) }-{\frac{7\,\sqrt{3}}{3}{\it Artanh} \left ({\frac{ \left ( 5\,{x}^{2}+6 \right ) \sqrt{3}}{6}{\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}}}} \right ) }+{\frac{2\,{x}^{2}+5}{6}\sqrt{{x}^{4}+5\,{x}^{2}+3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3/x^2*(x^4+5*x^2+3)^(3/2)+7/3*(x^4+5*x^2+3)^(1/2)+19/4*ln(x^2+5/2+(x^4+5*x^2+
3)^(1/2))-7/3*arctanh(1/6*(5*x^2+6)*3^(1/2)/(x^4+5*x^2+3)^(1/2))*3^(1/2)+1/6*(2*
x^2+5)*(x^4+5*x^2+3)^(1/2)

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Maxima [A]  time = 0.833406, size = 120, normalized size = 1.24 \[ -\frac{7}{3} \, \sqrt{3} \log \left (\frac{2 \, \sqrt{3} \sqrt{x^{4} + 5 \, x^{2} + 3}}{x^{2}} + \frac{6}{x^{2}} + 5\right ) + \frac{3}{2} \, \sqrt{x^{4} + 5 \, x^{2} + 3} - \frac{\sqrt{x^{4} + 5 \, x^{2} + 3}}{x^{2}} + \frac{19}{4} \, \log \left (2 \, x^{2} + 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} + 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7/3*sqrt(3)*log(2*sqrt(3)*sqrt(x^4 + 5*x^2 + 3)/x^2 + 6/x^2 + 5) + 3/2*sqrt(x^4
+ 5*x^2 + 3) - sqrt(x^4 + 5*x^2 + 3)/x^2 + 19/4*log(2*x^2 + 2*sqrt(x^4 + 5*x^2
+ 3) + 5)

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Fricas [A]  time = 0.280116, size = 419, normalized size = 4.32 \[ -\frac{8 \, \sqrt{3}{\left (12 \, x^{6} + 45 \, x^{4} - 2 \, x^{2} - 37\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} + 38 \,{\left (4 \, \sqrt{3}{\left (2 \, x^{4} + 5 \, x^{2}\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} - \sqrt{3}{\left (8 \, x^{6} + 40 \, x^{4} + 37 \, x^{2}\right )}\right )} \log \left (-2 \, x^{2} + 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} - 5\right ) + 56 \,{\left (8 \, x^{6} + 40 \, x^{4} + 37 \, x^{2} - 4 \,{\left (2 \, x^{4} + 5 \, x^{2}\right )} \sqrt{x^{4} + 5 \, x^{2} + 3}\right )} \log \left (\frac{6 \, x^{2} + \sqrt{3}{\left (2 \, x^{4} + 5 \, x^{2} + 6\right )} - 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (\sqrt{3} x^{2} + 3\right )}}{2 \, x^{4} - 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} x^{2} + 5 \, x^{2}}\right ) - \sqrt{3}{\left (96 \, x^{8} + 600 \, x^{6} + 728 \, x^{4} - 531 \, x^{2} - 480\right )}}{8 \,{\left (4 \, \sqrt{3}{\left (2 \, x^{4} + 5 \, x^{2}\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} - \sqrt{3}{\left (8 \, x^{6} + 40 \, x^{4} + 37 \, x^{2}\right )}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(8*sqrt(3)*(12*x^6 + 45*x^4 - 2*x^2 - 37)*sqrt(x^4 + 5*x^2 + 3) + 38*(4*sqr
t(3)*(2*x^4 + 5*x^2)*sqrt(x^4 + 5*x^2 + 3) - sqrt(3)*(8*x^6 + 40*x^4 + 37*x^2))*
log(-2*x^2 + 2*sqrt(x^4 + 5*x^2 + 3) - 5) + 56*(8*x^6 + 40*x^4 + 37*x^2 - 4*(2*x
^4 + 5*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)*(96*x^8 + 600*x^6 + 728*x^4 - 531*x^2 - 480))/(4*sqrt(3)*(2*x^
4 + 5*x^2)*sqrt(x^4 + 5*x^2 + 3) - sqrt(3)*(8*x^6 + 40*x^4 + 37*x^2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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