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

Optimal. Leaf size=84 \[ \frac{1}{3} \log \left (x^2+x+1\right )+\frac{1}{24} \log \left (2 x^2-x+2\right )-\frac{5}{2 x}-\frac{3 \log (x)}{4}+\frac{5}{12} \sqrt{\frac{5}{3}} \tan ^{-1}\left (\frac{1-4 x}{\sqrt{15}}\right )-\frac{10 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]

[Out]

-5/(2*x) + (5*Sqrt[5/3]*ArcTan[(1 - 4*x)/Sqrt[15]])/12 - (10*ArcTan[(1 + 2*x)/Sq
rt[3]])/(3*Sqrt[3]) - (3*Log[x])/4 + Log[1 + x + x^2]/3 + Log[2 - x + 2*x^2]/24

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Rubi [A]  time = 0.320216, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171 \[ \frac{1}{3} \log \left (x^2+x+1\right )+\frac{1}{24} \log \left (2 x^2-x+2\right )-\frac{5}{2 x}-\frac{3 \log (x)}{4}+\frac{5}{12} \sqrt{\frac{5}{3}} \tan ^{-1}\left (\frac{1-4 x}{\sqrt{15}}\right )-\frac{10 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

-5/(2*x) + (5*Sqrt[5/3]*ArcTan[(1 - 4*x)/Sqrt[15]])/12 - (10*ArcTan[(1 + 2*x)/Sq
rt[3]])/(3*Sqrt[3]) - (3*Log[x])/4 + Log[1 + x + x^2]/3 + Log[2 - x + 2*x^2]/24

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.0565308, size = 78, normalized size = 0.93 \[ -\frac{-24 x \log \left (x^2+x+1\right )-3 x \log \left (2 x^2-x+2\right )+54 x \log (x)+80 \sqrt{3} x \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )+10 \sqrt{15} x \tan ^{-1}\left (\frac{4 x-1}{\sqrt{15}}\right )+180}{72 x} \]

Antiderivative was successfully verified.

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

[Out]

-(180 + 80*Sqrt[3]*x*ArcTan[(1 + 2*x)/Sqrt[3]] + 10*Sqrt[15]*x*ArcTan[(-1 + 4*x)
/Sqrt[15]] + 54*x*Log[x] - 24*x*Log[1 + x + x^2] - 3*x*Log[2 - x + 2*x^2])/(72*x
)

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Maple [A]  time = 0.012, size = 65, normalized size = 0.8 \[{\frac{\ln \left ({x}^{2}+x+1 \right ) }{3}}-{\frac{10\,\sqrt{3}}{9}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }+{\frac{\ln \left ( 2\,{x}^{2}-x+2 \right ) }{24}}-{\frac{5\,\sqrt{15}}{36}\arctan \left ({\frac{ \left ( -1+4\,x \right ) \sqrt{15}}{15}} \right ) }-{\frac{5}{2\,x}}-{\frac{3\,\ln \left ( x \right ) }{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*ln(x^2+x+1)-10/9*arctan(1/3*(1+2*x)*3^(1/2))*3^(1/2)+1/24*ln(2*x^2-x+2)-5/36
*15^(1/2)*arctan(1/15*(-1+4*x)*15^(1/2))-5/2/x-3/4*ln(x)

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Maxima [A]  time = 0.893414, size = 86, normalized size = 1.02 \[ -\frac{5}{36} \, \sqrt{15} \arctan \left (\frac{1}{15} \, \sqrt{15}{\left (4 \, x - 1\right )}\right ) - \frac{10}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{5}{2 \, x} + \frac{1}{24} \, \log \left (2 \, x^{2} - x + 2\right ) + \frac{1}{3} \, \log \left (x^{2} + x + 1\right ) - \frac{3}{4} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/36*sqrt(15)*arctan(1/15*sqrt(15)*(4*x - 1)) - 10/9*sqrt(3)*arctan(1/3*sqrt(3)
*(2*x + 1)) - 5/2/x + 1/24*log(2*x^2 - x + 2) + 1/3*log(x^2 + x + 1) - 3/4*log(x
)

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Fricas [A]  time = 0.275648, size = 116, normalized size = 1.38 \[ -\frac{\sqrt{3}{\left (10 \, \sqrt{5} x \arctan \left (\frac{1}{15} \, \sqrt{5} \sqrt{3}{\left (4 \, x - 1\right )}\right ) - \sqrt{3} x \log \left (2 \, x^{2} - x + 2\right ) - 8 \, \sqrt{3} x \log \left (x^{2} + x + 1\right ) + 18 \, \sqrt{3} x \log \left (x\right ) + 80 \, x \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + 60 \, \sqrt{3}\right )}}{72 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/72*sqrt(3)*(10*sqrt(5)*x*arctan(1/15*sqrt(5)*sqrt(3)*(4*x - 1)) - sqrt(3)*x*l
og(2*x^2 - x + 2) - 8*sqrt(3)*x*log(x^2 + x + 1) + 18*sqrt(3)*x*log(x) + 80*x*ar
ctan(1/3*sqrt(3)*(2*x + 1)) + 60*sqrt(3))/x

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Sympy [A]  time = 0.942979, size = 87, normalized size = 1.04 \[ - \frac{3 \log{\left (x \right )}}{4} + \frac{\log{\left (x^{2} - \frac{x}{2} + 1 \right )}}{24} + \frac{\log{\left (x^{2} + x + 1 \right )}}{3} - \frac{5 \sqrt{15} \operatorname{atan}{\left (\frac{4 \sqrt{15} x}{15} - \frac{\sqrt{15}}{15} \right )}}{36} - \frac{10 \sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} + \frac{\sqrt{3}}{3} \right )}}{9} - \frac{5}{2 x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*log(x)/4 + log(x**2 - x/2 + 1)/24 + log(x**2 + x + 1)/3 - 5*sqrt(15)*atan(4*s
qrt(15)*x/15 - sqrt(15)/15)/36 - 10*sqrt(3)*atan(2*sqrt(3)*x/3 + sqrt(3)/3)/9 -
5/(2*x)

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GIAC/XCAS [A]  time = 0.262388, size = 88, normalized size = 1.05 \[ -\frac{5}{36} \, \sqrt{15} \arctan \left (\frac{1}{15} \, \sqrt{15}{\left (4 \, x - 1\right )}\right ) - \frac{10}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{5}{2 \, x} + \frac{1}{24} \,{\rm ln}\left (2 \, x^{2} - x + 2\right ) + \frac{1}{3} \,{\rm ln}\left (x^{2} + x + 1\right ) - \frac{3}{4} \,{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/36*sqrt(15)*arctan(1/15*sqrt(15)*(4*x - 1)) - 10/9*sqrt(3)*arctan(1/3*sqrt(3)
*(2*x + 1)) - 5/2/x + 1/24*ln(2*x^2 - x + 2) + 1/3*ln(x^2 + x + 1) - 3/4*ln(abs(
x))