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

Optimal. Leaf size=97 \[ \frac{x^4}{4}+\frac{x^3}{3}-\frac{3 x^2}{4}+\frac{1}{3} \log \left (x^2+x+1\right )-\frac{13}{48} \log \left (2 x^2-x+2\right )+\frac{5 x}{4}+\frac{1}{24} \sqrt{\frac{5}{3}} \tan ^{-1}\left (\frac{1-4 x}{\sqrt{15}}\right )-\frac{10 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]

[Out]

(5*x)/4 - (3*x^2)/4 + x^3/3 + x^4/4 + (Sqrt[5/3]*ArcTan[(1 - 4*x)/Sqrt[15]])/24
- (10*ArcTan[(1 + 2*x)/Sqrt[3]])/(3*Sqrt[3]) + Log[1 + x + x^2]/3 - (13*Log[2 -
x + 2*x^2])/48

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Rubi [A]  time = 0.265033, antiderivative size = 97, 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 \[ \frac{x^4}{4}+\frac{x^3}{3}-\frac{3 x^2}{4}+\frac{1}{3} \log \left (x^2+x+1\right )-\frac{13}{48} \log \left (2 x^2-x+2\right )+\frac{5 x}{4}+\frac{1}{24} \sqrt{\frac{5}{3}} \tan ^{-1}\left (\frac{1-4 x}{\sqrt{15}}\right )-\frac{10 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(5*x)/4 - (3*x^2)/4 + x^3/3 + x^4/4 + (Sqrt[5/3]*ArcTan[(1 - 4*x)/Sqrt[15]])/24
- (10*ArcTan[(1 + 2*x)/Sqrt[3]])/(3*Sqrt[3]) + Log[1 + x + x^2]/3 - (13*Log[2 -
x + 2*x^2])/48

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

[Out]

Timed out

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Mathematica [A]  time = 0.0566248, size = 83, normalized size = 0.86 \[ \frac{1}{144} \left (36 x^4+48 x^3-108 x^2+48 \log \left (x^2+x+1\right )-39 \log \left (2 x^2-x+2\right )+180 x-160 \sqrt{3} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )-2 \sqrt{15} \tan ^{-1}\left (\frac{4 x-1}{\sqrt{15}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(180*x - 108*x^2 + 48*x^3 + 36*x^4 - 160*Sqrt[3]*ArcTan[(1 + 2*x)/Sqrt[3]] - 2*S
qrt[15]*ArcTan[(-1 + 4*x)/Sqrt[15]] + 48*Log[1 + x + x^2] - 39*Log[2 - x + 2*x^2
])/144

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Maple [A]  time = 0.012, size = 74, normalized size = 0.8 \[{\frac{{x}^{4}}{4}}+{\frac{{x}^{3}}{3}}-{\frac{3\,{x}^{2}}{4}}+{\frac{5\,x}{4}}+{\frac{\ln \left ({x}^{2}+x+1 \right ) }{3}}-{\frac{10\,\sqrt{3}}{9}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{13\,\ln \left ( 2\,{x}^{2}-x+2 \right ) }{48}}-{\frac{\sqrt{15}}{72}\arctan \left ({\frac{ \left ( -1+4\,x \right ) \sqrt{15}}{15}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*x^4+1/3*x^3-3/4*x^2+5/4*x+1/3*ln(x^2+x+1)-10/9*arctan(1/3*(1+2*x)*3^(1/2))*3
^(1/2)-13/48*ln(2*x^2-x+2)-1/72*15^(1/2)*arctan(1/15*(-1+4*x)*15^(1/2))

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Maxima [A]  time = 0.904407, size = 99, normalized size = 1.02 \[ \frac{1}{4} \, x^{4} + \frac{1}{3} \, x^{3} - \frac{3}{4} \, x^{2} - \frac{1}{72} \, \sqrt{15} \arctan \left (\frac{1}{15} \, \sqrt{15}{\left (4 \, x - 1\right )}\right ) - \frac{10}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{5}{4} \, x - \frac{13}{48} \, \log \left (2 \, x^{2} - x + 2\right ) + \frac{1}{3} \, \log \left (x^{2} + x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*x^4 + 1/3*x^3 - 3/4*x^2 - 1/72*sqrt(15)*arctan(1/15*sqrt(15)*(4*x - 1)) - 10
/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + 1)) + 5/4*x - 13/48*log(2*x^2 - x + 2) + 1/
3*log(x^2 + x + 1)

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Fricas [A]  time = 0.273661, size = 122, normalized size = 1.26 \[ \frac{1}{144} \, \sqrt{3}{\left (4 \, \sqrt{3}{\left (3 \, x^{4} + 4 \, x^{3} - 9 \, x^{2} + 15 \, x\right )} - 2 \, \sqrt{5} \arctan \left (\frac{1}{15} \, \sqrt{5} \sqrt{3}{\left (4 \, x - 1\right )}\right ) - 13 \, \sqrt{3} \log \left (2 \, x^{2} - x + 2\right ) + 16 \, \sqrt{3} \log \left (x^{2} + x + 1\right ) - 160 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/144*sqrt(3)*(4*sqrt(3)*(3*x^4 + 4*x^3 - 9*x^2 + 15*x) - 2*sqrt(5)*arctan(1/15*
sqrt(5)*sqrt(3)*(4*x - 1)) - 13*sqrt(3)*log(2*x^2 - x + 2) + 16*sqrt(3)*log(x^2
+ x + 1) - 160*arctan(1/3*sqrt(3)*(2*x + 1)))

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Sympy [A]  time = 0.722517, size = 97, normalized size = 1. \[ \frac{x^{4}}{4} + \frac{x^{3}}{3} - \frac{3 x^{2}}{4} + \frac{5 x}{4} - \frac{13 \log{\left (x^{2} - \frac{x}{2} + 1 \right )}}{48} + \frac{\log{\left (x^{2} + x + 1 \right )}}{3} - \frac{\sqrt{15} \operatorname{atan}{\left (\frac{4 \sqrt{15} x}{15} - \frac{\sqrt{15}}{15} \right )}}{72} - \frac{10 \sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} + \frac{\sqrt{3}}{3} \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**4/4 + x**3/3 - 3*x**2/4 + 5*x/4 - 13*log(x**2 - x/2 + 1)/48 + log(x**2 + x +
1)/3 - sqrt(15)*atan(4*sqrt(15)*x/15 - sqrt(15)/15)/72 - 10*sqrt(3)*atan(2*sqrt(
3)*x/3 + sqrt(3)/3)/9

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GIAC/XCAS [A]  time = 0.263833, size = 99, normalized size = 1.02 \[ \frac{1}{4} \, x^{4} + \frac{1}{3} \, x^{3} - \frac{3}{4} \, x^{2} - \frac{1}{72} \, \sqrt{15} \arctan \left (\frac{1}{15} \, \sqrt{15}{\left (4 \, x - 1\right )}\right ) - \frac{10}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{5}{4} \, x - \frac{13}{48} \,{\rm ln}\left (2 \, x^{2} - x + 2\right ) + \frac{1}{3} \,{\rm ln}\left (x^{2} + x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*x^4 + 1/3*x^3 - 3/4*x^2 - 1/72*sqrt(15)*arctan(1/15*sqrt(15)*(4*x - 1)) - 10
/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + 1)) + 5/4*x - 13/48*ln(2*x^2 - x + 2) + 1/3
*ln(x^2 + x + 1)