3.264 \(\int \frac{x+10 x^2+6 x^3+x^4}{10+6 x+x^2} \, dx\)

Optimal. Leaf size=27 \[ \frac{x^3}{3}+\frac{1}{2} \log \left (x^2+6 x+10\right )-3 \tan ^{-1}(x+3) \]

[Out]

x^3/3 - 3*ArcTan[3 + x] + Log[10 + 6*x + x^2]/2

_______________________________________________________________________________________

Rubi [A]  time = 0.0486547, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{x^3}{3}+\frac{1}{2} \log \left (x^2+6 x+10\right )-3 \tan ^{-1}(x+3) \]

Antiderivative was successfully verified.

[In]  Int[(x + 10*x^2 + 6*x^3 + x^4)/(10 + 6*x + x^2),x]

[Out]

x^3/3 - 3*ArcTan[3 + x] + Log[10 + 6*x + x^2]/2

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 36.6503, size = 22, normalized size = 0.81 \[ \frac{x^{3}}{3} + \frac{\log{\left (x^{2} + 6 x + 10 \right )}}{2} - 3 \operatorname{atan}{\left (x + 3 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**4+6*x**3+10*x**2+x)/(x**2+6*x+10),x)

[Out]

x**3/3 + log(x**2 + 6*x + 10)/2 - 3*atan(x + 3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0115037, size = 27, normalized size = 1. \[ \frac{x^3}{3}+\frac{1}{2} \log \left (x^2+6 x+10\right )-3 \tan ^{-1}(x+3) \]

Antiderivative was successfully verified.

[In]  Integrate[(x + 10*x^2 + 6*x^3 + x^4)/(10 + 6*x + x^2),x]

[Out]

x^3/3 - 3*ArcTan[3 + x] + Log[10 + 6*x + x^2]/2

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 24, normalized size = 0.9 \[{\frac{{x}^{3}}{3}}-3\,\arctan \left ( 3+x \right ) +{\frac{\ln \left ({x}^{2}+6\,x+10 \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^4+6*x^3+10*x^2+x)/(x^2+6*x+10),x)

[Out]

1/3*x^3-3*arctan(3+x)+1/2*ln(x^2+6*x+10)

_______________________________________________________________________________________

Maxima [A]  time = 0.865046, size = 31, normalized size = 1.15 \[ \frac{1}{3} \, x^{3} - 3 \, \arctan \left (x + 3\right ) + \frac{1}{2} \, \log \left (x^{2} + 6 \, x + 10\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 + 6*x^3 + 10*x^2 + x)/(x^2 + 6*x + 10),x, algorithm="maxima")

[Out]

1/3*x^3 - 3*arctan(x + 3) + 1/2*log(x^2 + 6*x + 10)

_______________________________________________________________________________________

Fricas [A]  time = 0.273257, size = 31, normalized size = 1.15 \[ \frac{1}{3} \, x^{3} - 3 \, \arctan \left (x + 3\right ) + \frac{1}{2} \, \log \left (x^{2} + 6 \, x + 10\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 + 6*x^3 + 10*x^2 + x)/(x^2 + 6*x + 10),x, algorithm="fricas")

[Out]

1/3*x^3 - 3*arctan(x + 3) + 1/2*log(x^2 + 6*x + 10)

_______________________________________________________________________________________

Sympy [A]  time = 0.203127, size = 22, normalized size = 0.81 \[ \frac{x^{3}}{3} + \frac{\log{\left (x^{2} + 6 x + 10 \right )}}{2} - 3 \operatorname{atan}{\left (x + 3 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**4+6*x**3+10*x**2+x)/(x**2+6*x+10),x)

[Out]

x**3/3 + log(x**2 + 6*x + 10)/2 - 3*atan(x + 3)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.260273, size = 31, normalized size = 1.15 \[ \frac{1}{3} \, x^{3} - 3 \, \arctan \left (x + 3\right ) + \frac{1}{2} \,{\rm ln}\left (x^{2} + 6 \, x + 10\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 + 6*x^3 + 10*x^2 + x)/(x^2 + 6*x + 10),x, algorithm="giac")

[Out]

1/3*x^3 - 3*arctan(x + 3) + 1/2*ln(x^2 + 6*x + 10)