3.293 \(\int \frac{-3+x-2 x^3+x^4}{10-8 x+2 x^2} \, dx\)

Optimal. Leaf size=41 \[ \frac{x^3}{6}+\frac{x^2}{2}+\frac{3}{4} \log \left (x^2-4 x+5\right )+\frac{3 x}{2}+6 \tan ^{-1}(2-x) \]

[Out]

(3*x)/2 + x^2/2 + x^3/6 + 6*ArcTan[2 - x] + (3*Log[5 - 4*x + x^2])/4

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Rubi [A]  time = 0.0580116, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{x^3}{6}+\frac{x^2}{2}+\frac{3}{4} \log \left (x^2-4 x+5\right )+\frac{3 x}{2}+6 \tan ^{-1}(2-x) \]

Antiderivative was successfully verified.

[In]  Int[(-3 + x - 2*x^3 + x^4)/(10 - 8*x + 2*x^2),x]

[Out]

(3*x)/2 + x^2/2 + x^3/6 + 6*ArcTan[2 - x] + (3*Log[5 - 4*x + x^2])/4

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{x^{3}}{6} + \frac{x^{2}}{2} + \frac{3 \log{\left (2 x^{2} - 8 x + 10 \right )}}{4} - 6 \operatorname{atan}{\left (x - 2 \right )} + \int \frac{3}{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**4-2*x**3+x-3)/(2*x**2-8*x+10),x)

[Out]

x**3/6 + x**2/2 + 3*log(2*x**2 - 8*x + 10)/4 - 6*atan(x - 2) + Integral(3/2, x)

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Mathematica [A]  time = 0.0113821, size = 39, normalized size = 0.95 \[ \frac{1}{2} \left (\frac{x^3}{3}+x^2+\frac{3}{2} \log \left (x^2-4 x+5\right )+3 x+12 \tan ^{-1}(2-x)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(-3 + x - 2*x^3 + x^4)/(10 - 8*x + 2*x^2),x]

[Out]

(3*x + x^2 + x^3/3 + 12*ArcTan[2 - x] + (3*Log[5 - 4*x + x^2])/2)/2

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Maple [A]  time = 0.006, size = 32, normalized size = 0.8 \[{\frac{3\,x}{2}}+{\frac{{x}^{2}}{2}}+{\frac{{x}^{3}}{6}}-6\,\arctan \left ( x-2 \right ) +{\frac{3\,\ln \left ({x}^{2}-4\,x+5 \right ) }{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^4-2*x^3+x-3)/(2*x^2-8*x+10),x)

[Out]

3/2*x+1/2*x^2+1/6*x^3-6*arctan(x-2)+3/4*ln(x^2-4*x+5)

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Maxima [A]  time = 0.873511, size = 42, normalized size = 1.02 \[ \frac{1}{6} \, x^{3} + \frac{1}{2} \, x^{2} + \frac{3}{2} \, x - 6 \, \arctan \left (x - 2\right ) + \frac{3}{4} \, \log \left (x^{2} - 4 \, x + 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/2*(x^4 - 2*x^3 + x - 3)/(x^2 - 4*x + 5),x, algorithm="maxima")

[Out]

1/6*x^3 + 1/2*x^2 + 3/2*x - 6*arctan(x - 2) + 3/4*log(x^2 - 4*x + 5)

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Fricas [A]  time = 0.255419, size = 42, normalized size = 1.02 \[ \frac{1}{6} \, x^{3} + \frac{1}{2} \, x^{2} + \frac{3}{2} \, x - 6 \, \arctan \left (x - 2\right ) + \frac{3}{4} \, \log \left (x^{2} - 4 \, x + 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/2*(x^4 - 2*x^3 + x - 3)/(x^2 - 4*x + 5),x, algorithm="fricas")

[Out]

1/6*x^3 + 1/2*x^2 + 3/2*x - 6*arctan(x - 2) + 3/4*log(x^2 - 4*x + 5)

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Sympy [A]  time = 0.236565, size = 34, normalized size = 0.83 \[ \frac{x^{3}}{6} + \frac{x^{2}}{2} + \frac{3 x}{2} + \frac{3 \log{\left (x^{2} - 4 x + 5 \right )}}{4} - 6 \operatorname{atan}{\left (x - 2 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**4-2*x**3+x-3)/(2*x**2-8*x+10),x)

[Out]

x**3/6 + x**2/2 + 3*x/2 + 3*log(x**2 - 4*x + 5)/4 - 6*atan(x - 2)

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GIAC/XCAS [A]  time = 0.261423, size = 42, normalized size = 1.02 \[ \frac{1}{6} \, x^{3} + \frac{1}{2} \, x^{2} + \frac{3}{2} \, x - 6 \, \arctan \left (x - 2\right ) + \frac{3}{4} \,{\rm ln}\left (x^{2} - 4 \, x + 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/2*(x^4 - 2*x^3 + x - 3)/(x^2 - 4*x + 5),x, algorithm="giac")

[Out]

1/6*x^3 + 1/2*x^2 + 3/2*x - 6*arctan(x - 2) + 3/4*ln(x^2 - 4*x + 5)