3.2255 \(\int \frac{x^4}{4+4 x+x^2} \, dx\)

Optimal. Leaf size=29 \[ \frac{x^3}{3}-2 x^2+12 x-\frac{16}{x+2}-32 \log (x+2) \]

[Out]

12*x - 2*x^2 + x^3/3 - 16/(2 + x) - 32*Log[2 + x]

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Rubi [A]  time = 0.0362909, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{x^3}{3}-2 x^2+12 x-\frac{16}{x+2}-32 \log (x+2) \]

Antiderivative was successfully verified.

[In]  Int[x^4/(4 + 4*x + x^2),x]

[Out]

12*x - 2*x^2 + x^3/3 - 16/(2 + x) - 32*Log[2 + x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{x^{3}}{3} + 12 x - 32 \log{\left (x + 2 \right )} - 4 \int x\, dx - \frac{16}{x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**3/3 + 12*x - 32*log(x + 2) - 4*Integral(x, x) - 16/(x + 2)

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Mathematica [A]  time = 0.0177923, size = 30, normalized size = 1.03 \[ \frac{1}{3} \left (x^3-6 x^2+36 x-\frac{48}{x+2}-96 \log (x+2)+104\right ) \]

Antiderivative was successfully verified.

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

[Out]

(104 + 36*x - 6*x^2 + x^3 - 48/(2 + x) - 96*Log[2 + x])/3

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Maple [A]  time = 0.009, size = 28, normalized size = 1. \[ 12\,x-2\,{x}^{2}+{\frac{{x}^{3}}{3}}-16\, \left ( 2+x \right ) ^{-1}-32\,\ln \left ( 2+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(x^2+4*x+4),x)

[Out]

12*x-2*x^2+1/3*x^3-16/(2+x)-32*ln(2+x)

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Maxima [A]  time = 0.681258, size = 36, normalized size = 1.24 \[ \frac{1}{3} \, x^{3} - 2 \, x^{2} + 12 \, x - \frac{16}{x + 2} - 32 \, \log \left (x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*x^3 - 2*x^2 + 12*x - 16/(x + 2) - 32*log(x + 2)

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Fricas [A]  time = 0.199066, size = 46, normalized size = 1.59 \[ \frac{x^{4} - 4 \, x^{3} + 24 \, x^{2} - 96 \,{\left (x + 2\right )} \log \left (x + 2\right ) + 72 \, x - 48}{3 \,{\left (x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(x^4 - 4*x^3 + 24*x^2 - 96*(x + 2)*log(x + 2) + 72*x - 48)/(x + 2)

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Sympy [A]  time = 0.148384, size = 24, normalized size = 0.83 \[ \frac{x^{3}}{3} - 2 x^{2} + 12 x - 32 \log{\left (x + 2 \right )} - \frac{16}{x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(x**2+4*x+4),x)

[Out]

x**3/3 - 2*x**2 + 12*x - 32*log(x + 2) - 16/(x + 2)

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GIAC/XCAS [A]  time = 0.204577, size = 38, normalized size = 1.31 \[ \frac{1}{3} \, x^{3} - 2 \, x^{2} + 12 \, x - \frac{16}{x + 2} - 32 \,{\rm ln}\left ({\left | x + 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*x^3 - 2*x^2 + 12*x - 16/(x + 2) - 32*ln(abs(x + 2))