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

Optimal. Leaf size=77 \[ \frac{x^2}{2}-\log \left (x^2+x+1\right )+\frac{1}{4} \log \left (2 x^2-x+2\right )+x+\frac{1}{6} \sqrt{\frac{5}{3}} \tan ^{-1}\left (\frac{1-4 x}{\sqrt{15}}\right )+\frac{2 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]

[Out]

x + x^2/2 + (Sqrt[5/3]*ArcTan[(1 - 4*x)/Sqrt[15]])/6 + (2*ArcTan[(1 + 2*x)/Sqrt[
3]])/(3*Sqrt[3]) - Log[1 + x + x^2] + Log[2 - x + 2*x^2]/4

_______________________________________________________________________________________

Rubi [A]  time = 0.236505, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{x^2}{2}-\log \left (x^2+x+1\right )+\frac{1}{4} \log \left (2 x^2-x+2\right )+x+\frac{1}{6} \sqrt{\frac{5}{3}} \tan ^{-1}\left (\frac{1-4 x}{\sqrt{15}}\right )+\frac{2 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

x + x^2/2 + (Sqrt[5/3]*ArcTan[(1 - 4*x)/Sqrt[15]])/6 + (2*ArcTan[(1 + 2*x)/Sqrt[
3]])/(3*Sqrt[3]) - Log[1 + x + x^2] + Log[2 - x + 2*x^2]/4

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 0.049007, size = 72, normalized size = 0.94 \[ \frac{1}{36} \left (9 \left (-4 \log \left (x^2+x+1\right )+\log \left (2 x^2-x+2\right )+2 x (x+2)\right )+8 \sqrt{3} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )-2 \sqrt{15} \tan ^{-1}\left (\frac{4 x-1}{\sqrt{15}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(8*Sqrt[3]*ArcTan[(1 + 2*x)/Sqrt[3]] - 2*Sqrt[15]*ArcTan[(-1 + 4*x)/Sqrt[15]] +
9*(2*x*(2 + x) - 4*Log[1 + x + x^2] + Log[2 - x + 2*x^2]))/36

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 62, normalized size = 0.8 \[ x+{\frac{{x}^{2}}{2}}-\ln \left ({x}^{2}+x+1 \right ) +{\frac{2\,\sqrt{3}}{9}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }+{\frac{\ln \left ( 2\,{x}^{2}-x+2 \right ) }{4}}-{\frac{\sqrt{15}}{18}\arctan \left ({\frac{ \left ( -1+4\,x \right ) \sqrt{15}}{15}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x+1/2*x^2-ln(x^2+x+1)+2/9*arctan(1/3*(1+2*x)*3^(1/2))*3^(1/2)+1/4*ln(2*x^2-x+2)-
1/18*15^(1/2)*arctan(1/15*(-1+4*x)*15^(1/2))

_______________________________________________________________________________________

Maxima [A]  time = 0.914693, size = 82, normalized size = 1.06 \[ \frac{1}{2} \, x^{2} - \frac{1}{18} \, \sqrt{15} \arctan \left (\frac{1}{15} \, \sqrt{15}{\left (4 \, x - 1\right )}\right ) + \frac{2}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + x + \frac{1}{4} \, \log \left (2 \, x^{2} - x + 2\right ) - \log \left (x^{2} + x + 1\right ) \]

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, algorithm="maxima")

[Out]

1/2*x^2 - 1/18*sqrt(15)*arctan(1/15*sqrt(15)*(4*x - 1)) + 2/9*sqrt(3)*arctan(1/3
*sqrt(3)*(2*x + 1)) + x + 1/4*log(2*x^2 - x + 2) - log(x^2 + x + 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.275754, size = 105, normalized size = 1.36 \[ \frac{1}{36} \, \sqrt{3}{\left (6 \, \sqrt{3}{\left (x^{2} + 2 \, x\right )} - 2 \, \sqrt{5} \arctan \left (\frac{1}{15} \, \sqrt{5} \sqrt{3}{\left (4 \, x - 1\right )}\right ) + 3 \, \sqrt{3} \log \left (2 \, x^{2} - x + 2\right ) - 12 \, \sqrt{3} \log \left (x^{2} + x + 1\right ) + 8 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right )\right )} \]

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, algorithm="fricas")

[Out]

1/36*sqrt(3)*(6*sqrt(3)*(x^2 + 2*x) - 2*sqrt(5)*arctan(1/15*sqrt(5)*sqrt(3)*(4*x
- 1)) + 3*sqrt(3)*log(2*x^2 - x + 2) - 12*sqrt(3)*log(x^2 + x + 1) + 8*arctan(1
/3*sqrt(3)*(2*x + 1)))

_______________________________________________________________________________________

Sympy [A]  time = 0.806577, size = 78, normalized size = 1.01 \[ \frac{x^{2}}{2} + x + \frac{\log{\left (x^{2} - \frac{x}{2} + 1 \right )}}{4} - \log{\left (x^{2} + x + 1 \right )} - \frac{\sqrt{15} \operatorname{atan}{\left (\frac{4 \sqrt{15} x}{15} - \frac{\sqrt{15}}{15} \right )}}{18} + \frac{2 \sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} + \frac{\sqrt{3}}{3} \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**2/2 + x + log(x**2 - x/2 + 1)/4 - log(x**2 + x + 1) - sqrt(15)*atan(4*sqrt(15
)*x/15 - sqrt(15)/15)/18 + 2*sqrt(3)*atan(2*sqrt(3)*x/3 + sqrt(3)/3)/9

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.260861, size = 82, normalized size = 1.06 \[ \frac{1}{2} \, x^{2} - \frac{1}{18} \, \sqrt{15} \arctan \left (\frac{1}{15} \, \sqrt{15}{\left (4 \, x - 1\right )}\right ) + \frac{2}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + x + \frac{1}{4} \,{\rm ln}\left (2 \, x^{2} - x + 2\right ) -{\rm ln}\left (x^{2} + x + 1\right ) \]

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, algorithm="giac")

[Out]

1/2*x^2 - 1/18*sqrt(15)*arctan(1/15*sqrt(15)*(4*x - 1)) + 2/9*sqrt(3)*arctan(1/3
*sqrt(3)*(2*x + 1)) + x + 1/4*ln(2*x^2 - x + 2) - ln(x^2 + x + 1)