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