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

Optimal. Leaf size=90 \[ \frac{x^3}{3}+\frac{x^2}{2}+\frac{2}{3} \log \left (x^2+x+1\right )-\frac{1}{24} \log \left (2 x^2-x+2\right )-\frac{3 x}{2}+\frac{5}{12} \sqrt{\frac{5}{3}} \tan ^{-1}\left (\frac{1-4 x}{\sqrt{15}}\right )+\frac{8 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]

[Out]

(-3*x)/2 + x^2/2 + x^3/3 + (5*Sqrt[5/3]*ArcTan[(1 - 4*x)/Sqrt[15]])/12 + (8*ArcT
an[(1 + 2*x)/Sqrt[3]])/(3*Sqrt[3]) + (2*Log[1 + x + x^2])/3 - Log[2 - x + 2*x^2]
/24

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Rubi [A]  time = 0.236407, antiderivative size = 90, 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 \[ \frac{x^3}{3}+\frac{x^2}{2}+\frac{2}{3} \log \left (x^2+x+1\right )-\frac{1}{24} \log \left (2 x^2-x+2\right )-\frac{3 x}{2}+\frac{5}{12} \sqrt{\frac{5}{3}} \tan ^{-1}\left (\frac{1-4 x}{\sqrt{15}}\right )+\frac{8 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(-3*x)/2 + x^2/2 + x^3/3 + (5*Sqrt[5/3]*ArcTan[(1 - 4*x)/Sqrt[15]])/12 + (8*ArcT
an[(1 + 2*x)/Sqrt[3]])/(3*Sqrt[3]) + (2*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(x**3*(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 = 0.0406212, size = 78, normalized size = 0.87 \[ \frac{1}{72} \left (24 x^3+36 x^2+48 \log \left (x^2+x+1\right )-3 \log \left (2 x^2-x+2\right )-108 x+64 \sqrt{3} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )-10 \sqrt{15} \tan ^{-1}\left (\frac{4 x-1}{\sqrt{15}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-108*x + 36*x^2 + 24*x^3 + 64*Sqrt[3]*ArcTan[(1 + 2*x)/Sqrt[3]] - 10*Sqrt[15]*A
rcTan[(-1 + 4*x)/Sqrt[15]] + 48*Log[1 + x + x^2] - 3*Log[2 - x + 2*x^2])/72

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Maple [A]  time = 0.009, size = 69, normalized size = 0.8 \[{\frac{{x}^{3}}{3}}+{\frac{{x}^{2}}{2}}-{\frac{3\,x}{2}}+{\frac{2\,\ln \left ({x}^{2}+x+1 \right ) }{3}}+{\frac{8\,\sqrt{3}}{9}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }-{\frac{\ln \left ( 2\,{x}^{2}-x+2 \right ) }{24}}-{\frac{5\,\sqrt{15}}{36}\arctan \left ({\frac{ \left ( -1+4\,x \right ) \sqrt{15}}{15}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*x^3+1/2*x^2-3/2*x+2/3*ln(x^2+x+1)+8/9*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))

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Maxima [A]  time = 0.914183, size = 92, normalized size = 1.02 \[ \frac{1}{3} \, x^{3} + \frac{1}{2} \, x^{2} - \frac{5}{36} \, \sqrt{15} \arctan \left (\frac{1}{15} \, \sqrt{15}{\left (4 \, x - 1\right )}\right ) + \frac{8}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{3}{2} \, x - \frac{1}{24} \, \log \left (2 \, x^{2} - x + 2\right ) + \frac{2}{3} \, \log \left (x^{2} + x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*x^3 + 1/2*x^2 - 5/36*sqrt(15)*arctan(1/15*sqrt(15)*(4*x - 1)) + 8/9*sqrt(3)*
arctan(1/3*sqrt(3)*(2*x + 1)) - 3/2*x - 1/24*log(2*x^2 - x + 2) + 2/3*log(x^2 +
x + 1)

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Fricas [A]  time = 0.274183, size = 115, normalized size = 1.28 \[ \frac{1}{72} \, \sqrt{3}{\left (4 \, \sqrt{3}{\left (2 \, x^{3} + 3 \, x^{2} - 9 \, x\right )} - 10 \, \sqrt{5} \arctan \left (\frac{1}{15} \, \sqrt{5} \sqrt{3}{\left (4 \, x - 1\right )}\right ) - \sqrt{3} \log \left (2 \, x^{2} - x + 2\right ) + 16 \, \sqrt{3} \log \left (x^{2} + x + 1\right ) + 64 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/72*sqrt(3)*(4*sqrt(3)*(2*x^3 + 3*x^2 - 9*x) - 10*sqrt(5)*arctan(1/15*sqrt(5)*s
qrt(3)*(4*x - 1)) - sqrt(3)*log(2*x^2 - x + 2) + 16*sqrt(3)*log(x^2 + x + 1) + 6
4*arctan(1/3*sqrt(3)*(2*x + 1)))

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Sympy [A]  time = 0.684268, size = 92, normalized size = 1.02 \[ \frac{x^{3}}{3} + \frac{x^{2}}{2} - \frac{3 x}{2} - \frac{\log{\left (x^{2} - \frac{x}{2} + 1 \right )}}{24} + \frac{2 \log{\left (x^{2} + x + 1 \right )}}{3} - \frac{5 \sqrt{15} \operatorname{atan}{\left (\frac{4 \sqrt{15} x}{15} - \frac{\sqrt{15}}{15} \right )}}{36} + \frac{8 \sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} + \frac{\sqrt{3}}{3} \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**3/3 + x**2/2 - 3*x/2 - log(x**2 - x/2 + 1)/24 + 2*log(x**2 + x + 1)/3 - 5*sqr
t(15)*atan(4*sqrt(15)*x/15 - sqrt(15)/15)/36 + 8*sqrt(3)*atan(2*sqrt(3)*x/3 + sq
rt(3)/3)/9

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GIAC/XCAS [A]  time = 0.261816, size = 92, normalized size = 1.02 \[ \frac{1}{3} \, x^{3} + \frac{1}{2} \, x^{2} - \frac{5}{36} \, \sqrt{15} \arctan \left (\frac{1}{15} \, \sqrt{15}{\left (4 \, x - 1\right )}\right ) + \frac{8}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{3}{2} \, x - \frac{1}{24} \,{\rm ln}\left (2 \, x^{2} - x + 2\right ) + \frac{2}{3} \,{\rm ln}\left (x^{2} + x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*x^3 + 1/2*x^2 - 5/36*sqrt(15)*arctan(1/15*sqrt(15)*(4*x - 1)) + 8/9*sqrt(3)*
arctan(1/3*sqrt(3)*(2*x + 1)) - 3/2*x - 1/24*ln(2*x^2 - x + 2) + 2/3*ln(x^2 + x
+ 1)