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

Optimal. Leaf size=91 \[ -\frac{5}{4 x^2}+\frac{2}{3} \log \left (x^2+x+1\right )+\frac{13}{48} \log \left (2 x^2-x+2\right )+\frac{3}{4 x}-\frac{15 \log (x)}{8}+\frac{1}{24} \sqrt{\frac{5}{3}} \tan ^{-1}\left (\frac{1-4 x}{\sqrt{15}}\right )+\frac{8 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]

[Out]

-5/(4*x^2) + 3/(4*x) + (Sqrt[5/3]*ArcTan[(1 - 4*x)/Sqrt[15]])/24 + (8*ArcTan[(1
+ 2*x)/Sqrt[3]])/(3*Sqrt[3]) - (15*Log[x])/8 + (2*Log[1 + x + x^2])/3 + (13*Log[
2 - x + 2*x^2])/48

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Rubi [A]  time = 0.330969, antiderivative size = 91, 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 \[ -\frac{5}{4 x^2}+\frac{2}{3} \log \left (x^2+x+1\right )+\frac{13}{48} \log \left (2 x^2-x+2\right )+\frac{3}{4 x}-\frac{15 \log (x)}{8}+\frac{1}{24} \sqrt{\frac{5}{3}} \tan ^{-1}\left (\frac{1-4 x}{\sqrt{15}}\right )+\frac{8 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

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

[Out]

Timed out

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Mathematica [A]  time = 0.102335, size = 82, normalized size = 0.9 \[ \frac{1}{144} \left (3 \left (-\frac{60}{x^2}+32 \log \left (x^2+x+1\right )+13 \log \left (2 x^2-x+2\right )+\frac{36}{x}-90 \log (x)\right )+128 \sqrt{3} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )-2 \sqrt{15} \tan ^{-1}\left (\frac{4 x-1}{\sqrt{15}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(128*Sqrt[3]*ArcTan[(1 + 2*x)/Sqrt[3]] - 2*Sqrt[15]*ArcTan[(-1 + 4*x)/Sqrt[15]]
+ 3*(-60/x^2 + 36/x - 90*Log[x] + 32*Log[1 + x + x^2] + 13*Log[2 - x + 2*x^2]))/
144

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Maple [A]  time = 0.012, size = 70, normalized size = 0.8 \[{\frac{2\,\ln \left ({x}^{2}+x+1 \right ) }{3}}+{\frac{8\,\sqrt{3}}{9}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }+{\frac{13\,\ln \left ( 2\,{x}^{2}-x+2 \right ) }{48}}-{\frac{\sqrt{15}}{72}\arctan \left ({\frac{ \left ( -1+4\,x \right ) \sqrt{15}}{15}} \right ) }-{\frac{5}{4\,{x}^{2}}}+{\frac{3}{4\,x}}-{\frac{15\,\ln \left ( x \right ) }{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.918459, size = 93, normalized size = 1.02 \[ -\frac{1}{72} \, \sqrt{15} \arctan \left (\frac{1}{15} \, \sqrt{15}{\left (4 \, x - 1\right )}\right ) + \frac{8}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{3 \, x - 5}{4 \, x^{2}} + \frac{13}{48} \, \log \left (2 \, x^{2} - x + 2\right ) + \frac{2}{3} \, \log \left (x^{2} + x + 1\right ) - \frac{15}{8} \, \log \left (x\right ) \]

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^3),x, algorithm="maxima")

[Out]

-1/72*sqrt(15)*arctan(1/15*sqrt(15)*(4*x - 1)) + 8/9*sqrt(3)*arctan(1/3*sqrt(3)*
(2*x + 1)) + 1/4*(3*x - 5)/x^2 + 13/48*log(2*x^2 - x + 2) + 2/3*log(x^2 + x + 1)
- 15/8*log(x)

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Fricas [A]  time = 0.276062, size = 136, normalized size = 1.49 \[ -\frac{\sqrt{3}{\left (2 \, \sqrt{5} x^{2} \arctan \left (\frac{1}{15} \, \sqrt{5} \sqrt{3}{\left (4 \, x - 1\right )}\right ) - 13 \, \sqrt{3} x^{2} \log \left (2 \, x^{2} - x + 2\right ) - 32 \, \sqrt{3} x^{2} \log \left (x^{2} + x + 1\right ) + 90 \, \sqrt{3} x^{2} \log \left (x\right ) - 128 \, x^{2} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - 12 \, \sqrt{3}{\left (3 \, x - 5\right )}\right )}}{144 \, x^{2}} \]

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^3),x, algorithm="fricas")

[Out]

-1/144*sqrt(3)*(2*sqrt(5)*x^2*arctan(1/15*sqrt(5)*sqrt(3)*(4*x - 1)) - 13*sqrt(3
)*x^2*log(2*x^2 - x + 2) - 32*sqrt(3)*x^2*log(x^2 + x + 1) + 90*sqrt(3)*x^2*log(
x) - 128*x^2*arctan(1/3*sqrt(3)*(2*x + 1)) - 12*sqrt(3)*(3*x - 5))/x^2

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Sympy [A]  time = 1.01566, size = 94, normalized size = 1.03 \[ - \frac{15 \log{\left (x \right )}}{8} + \frac{13 \log{\left (x^{2} - \frac{x}{2} + 1 \right )}}{48} + \frac{2 \log{\left (x^{2} + x + 1 \right )}}{3} - \frac{\sqrt{15} \operatorname{atan}{\left (\frac{4 \sqrt{15} x}{15} - \frac{\sqrt{15}}{15} \right )}}{72} + \frac{8 \sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} + \frac{\sqrt{3}}{3} \right )}}{9} + \frac{3 x - 5}{4 x^{2}} \]

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)

[Out]

-15*log(x)/8 + 13*log(x**2 - x/2 + 1)/48 + 2*log(x**2 + x + 1)/3 - sqrt(15)*atan
(4*sqrt(15)*x/15 - sqrt(15)/15)/72 + 8*sqrt(3)*atan(2*sqrt(3)*x/3 + sqrt(3)/3)/9
+ (3*x - 5)/(4*x**2)

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GIAC/XCAS [A]  time = 0.263831, size = 95, normalized size = 1.04 \[ -\frac{1}{72} \, \sqrt{15} \arctan \left (\frac{1}{15} \, \sqrt{15}{\left (4 \, x - 1\right )}\right ) + \frac{8}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{3 \, x - 5}{4 \, x^{2}} + \frac{13}{48} \,{\rm ln}\left (2 \, x^{2} - x + 2\right ) + \frac{2}{3} \,{\rm ln}\left (x^{2} + x + 1\right ) - \frac{15}{8} \,{\rm ln}\left ({\left | x \right |}\right ) \]

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^3),x, algorithm="giac")

[Out]

-1/72*sqrt(15)*arctan(1/15*sqrt(15)*(4*x - 1)) + 8/9*sqrt(3)*arctan(1/3*sqrt(3)*
(2*x + 1)) + 1/4*(3*x - 5)/x^2 + 13/48*ln(2*x^2 - x + 2) + 2/3*ln(x^2 + x + 1) -
15/8*ln(abs(x))