Optimal. Leaf size=51 \[ \log (x+2) (d-2 e+4 f-8 g)+x (e-4 f+12 g)+\frac{1}{2} (x+2)^2 (f-6 g)+\frac{1}{3} g (x+2)^3 \]
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Rubi [A] time = 0.127868, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.049 \[ \log (x+2) (d-2 e+4 f-8 g)+x (e-4 f+12 g)+\frac{1}{2} (x+2)^2 (f-6 g)+\frac{1}{3} g (x+2)^3 \]
Antiderivative was successfully verified.
[In] Int[((2 - x - 2*x^2 + x^3)*(d + e*x + f*x^2 + g*x^3))/(4 - 5*x^2 + x^4),x]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - 4 f x + 12 g x + \frac{g \left (x + 2\right )^{3}}{3} + \left (\frac{f}{2} - 3 g\right ) \left (x + 2\right )^{2} + \left (d - 2 e + 4 f - 8 g\right ) \log{\left (x + 2 \right )} + \int e\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((x**3-2*x**2-x+2)*(g*x**3+f*x**2+e*x+d)/(x**4-5*x**2+4),x)
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Mathematica [A] time = 0.0432345, size = 45, normalized size = 0.88 \[ \log (x+2) (d-2 e+4 f-8 g)+\frac{1}{6} (x+2) \left (6 e+3 f (x-6)+2 g \left (x^2-5 x+22\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[((2 - x - 2*x^2 + x^3)*(d + e*x + f*x^2 + g*x^3))/(4 - 5*x^2 + x^4),x]
[Out]
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Maple [A] time = 0.004, size = 58, normalized size = 1.1 \[{\frac{g{x}^{3}}{3}}+{\frac{f{x}^{2}}{2}}-g{x}^{2}+ex-2\,fx+4\,gx+\ln \left ( 2+x \right ) d-2\,\ln \left ( 2+x \right ) e+4\,\ln \left ( 2+x \right ) f-8\,\ln \left ( 2+x \right ) g \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((x^3-2*x^2-x+2)*(g*x^3+f*x^2+e*x+d)/(x^4-5*x^2+4),x)
[Out]
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Maxima [A] time = 0.693303, size = 58, normalized size = 1.14 \[ \frac{1}{3} \, g x^{3} + \frac{1}{2} \,{\left (f - 2 \, g\right )} x^{2} +{\left (e - 2 \, f + 4 \, g\right )} x +{\left (d - 2 \, e + 4 \, f - 8 \, g\right )} \log \left (x + 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((g*x^3 + f*x^2 + e*x + d)*(x^3 - 2*x^2 - x + 2)/(x^4 - 5*x^2 + 4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.26788, size = 58, normalized size = 1.14 \[ \frac{1}{3} \, g x^{3} + \frac{1}{2} \,{\left (f - 2 \, g\right )} x^{2} +{\left (e - 2 \, f + 4 \, g\right )} x +{\left (d - 2 \, e + 4 \, f - 8 \, g\right )} \log \left (x + 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((g*x^3 + f*x^2 + e*x + d)*(x^3 - 2*x^2 - x + 2)/(x^4 - 5*x^2 + 4),x, algorithm="fricas")
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Sympy [A] time = 1.21559, size = 41, normalized size = 0.8 \[ \frac{g x^{3}}{3} + x^{2} \left (\frac{f}{2} - g\right ) + x \left (e - 2 f + 4 g\right ) + \left (d - 2 e + 4 f - 8 g\right ) \log{\left (x + 2 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x**3-2*x**2-x+2)*(g*x**3+f*x**2+e*x+d)/(x**4-5*x**2+4),x)
[Out]
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GIAC/XCAS [A] time = 0.289, size = 66, normalized size = 1.29 \[ \frac{1}{3} \, g x^{3} + \frac{1}{2} \, f x^{2} - g x^{2} - 2 \, f x + 4 \, g x + x e +{\left (d + 4 \, f - 8 \, g - 2 \, e\right )}{\rm ln}\left ({\left | x + 2 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((g*x^3 + f*x^2 + e*x + d)*(x^3 - 2*x^2 - x + 2)/(x^4 - 5*x^2 + 4),x, algorithm="giac")
[Out]