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

Optimal. Leaf size=94 \[ \frac{1}{8} \sqrt{x^4+5 x^2+3} \left (6 x^2+23\right )+\frac{1}{16} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right )-\sqrt{3} \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right ) \]

[Out]

((23 + 6*x^2)*Sqrt[3 + 5*x^2 + x^4])/8 + ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5*x^2 +
x^4])]/16 - Sqrt[3]*ArcTanh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4])]

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Rubi [A]  time = 0.1996, antiderivative size = 94, 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 \[ \frac{1}{8} \sqrt{x^4+5 x^2+3} \left (6 x^2+23\right )+\frac{1}{16} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right )-\sqrt{3} \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((23 + 6*x^2)*Sqrt[3 + 5*x^2 + x^4])/8 + ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5*x^2 +
x^4])]/16 - Sqrt[3]*ArcTanh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4])]

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Rubi in Sympy [A]  time = 21.1851, size = 82, normalized size = 0.87 \[ \frac{\left (3 x^{2} + \frac{23}{2}\right ) \sqrt{x^{4} + 5 x^{2} + 3}}{4} + \frac{\operatorname{atanh}{\left (\frac{2 x^{2} + 5}{2 \sqrt{x^{4} + 5 x^{2} + 3}} \right )}}{16} - \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (5 x^{2} + 6\right )}{6 \sqrt{x^{4} + 5 x^{2} + 3}} \right )} \]

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,x)

[Out]

(3*x**2 + 23/2)*sqrt(x**4 + 5*x**2 + 3)/4 + atanh((2*x**2 + 5)/(2*sqrt(x**4 + 5*
x**2 + 3)))/16 - sqrt(3)*atanh(sqrt(3)*(5*x**2 + 6)/(6*sqrt(x**4 + 5*x**2 + 3)))

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Mathematica [A]  time = 0.126806, size = 100, normalized size = 1.06 \[ \sqrt{3} \log \left (x^2\right )+\frac{1}{8} \sqrt{x^4+5 x^2+3} \left (6 x^2+23\right )+\frac{1}{16} \log \left (2 x^2+2 \sqrt{x^4+5 x^2+3}+5\right )-\sqrt{3} \log \left (5 x^2+2 \sqrt{3} \sqrt{x^4+5 x^2+3}+6\right ) \]

Antiderivative was successfully verified.

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

[Out]

((23 + 6*x^2)*Sqrt[3 + 5*x^2 + x^4])/8 + Sqrt[3]*Log[x^2] + Log[5 + 2*x^2 + 2*Sq
rt[3 + 5*x^2 + x^4]]/16 - Sqrt[3]*Log[6 + 5*x^2 + 2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4
]]

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Maple [A]  time = 0.018, size = 85, normalized size = 0.9 \[ \sqrt{{x}^{4}+5\,{x}^{2}+3}+{\frac{1}{16}\ln \left ({x}^{2}+{\frac{5}{2}}+\sqrt{{x}^{4}+5\,{x}^{2}+3} \right ) }-{\it Artanh} \left ({\frac{ \left ( 5\,{x}^{2}+6 \right ) \sqrt{3}}{6}{\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}}}} \right ) \sqrt{3}+{\frac{6\,{x}^{2}+15}{8}\sqrt{{x}^{4}+5\,{x}^{2}+3}} \]

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,x)

[Out]

(x^4+5*x^2+3)^(1/2)+1/16*ln(x^2+5/2+(x^4+5*x^2+3)^(1/2))-arctanh(1/6*(5*x^2+6)*3
^(1/2)/(x^4+5*x^2+3)^(1/2))*3^(1/2)+3/8*(2*x^2+5)*(x^4+5*x^2+3)^(1/2)

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Maxima [A]  time = 0.859078, size = 120, normalized size = 1.28 \[ \frac{3}{4} \, \sqrt{x^{4} + 5 \, x^{2} + 3} x^{2} - \sqrt{3} \log \left (\frac{2 \, \sqrt{3} \sqrt{x^{4} + 5 \, x^{2} + 3}}{x^{2}} + \frac{6}{x^{2}} + 5\right ) + \frac{23}{8} \, \sqrt{x^{4} + 5 \, x^{2} + 3} + \frac{1}{16} \, \log \left (2 \, x^{2} + 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} + 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/4*sqrt(x^4 + 5*x^2 + 3)*x^2 - sqrt(3)*log(2*sqrt(3)*sqrt(x^4 + 5*x^2 + 3)/x^2
+ 6/x^2 + 5) + 23/8*sqrt(x^4 + 5*x^2 + 3) + 1/16*log(2*x^2 + 2*sqrt(x^4 + 5*x^2
+ 3) + 5)

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Fricas [A]  time = 0.266154, size = 356, normalized size = 3.79 \[ -\frac{384 \, x^{8} + 4352 \, x^{6} + 15752 \, x^{4} + 19496 \, x^{2} + 4 \,{\left (8 \, x^{4} + 40 \, x^{2} - 4 \, \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (2 \, x^{2} + 5\right )} + 37\right )} \log \left (-2 \, x^{2} + 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} - 5\right ) + 64 \,{\left (4 \, \sqrt{3} \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (2 \, x^{2} + 5\right )} - \sqrt{3}{\left (8 \, x^{4} + 40 \, x^{2} + 37\right )}\right )} \log \left (\frac{2 \, x^{4} + 2 \, \sqrt{3} x^{2} + 5 \, x^{2} - 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (x^{2} + \sqrt{3}\right )} + 6}{2 \, x^{4} - 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} x^{2} + 5 \, x^{2}}\right ) - 4 \,{\left (96 \, x^{6} + 848 \, x^{4} + 1974 \, x^{2} + 927\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} + 5305}{64 \,{\left (8 \, x^{4} + 40 \, x^{2} - 4 \, \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (2 \, x^{2} + 5\right )} + 37\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/64*(384*x^8 + 4352*x^6 + 15752*x^4 + 19496*x^2 + 4*(8*x^4 + 40*x^2 - 4*sqrt(x
^4 + 5*x^2 + 3)*(2*x^2 + 5) + 37)*log(-2*x^2 + 2*sqrt(x^4 + 5*x^2 + 3) - 5) + 64
*(4*sqrt(3)*sqrt(x^4 + 5*x^2 + 3)*(2*x^2 + 5) - sqrt(3)*(8*x^4 + 40*x^2 + 37))*l
og((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)) - 4*(96*x^6 + 848*x^4 + 1974*x^2
+ 927)*sqrt(x^4 + 5*x^2 + 3) + 5305)/(8*x^4 + 40*x^2 - 4*sqrt(x^4 + 5*x^2 + 3)*
(2*x^2 + 5) + 37)

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

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,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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