3.2253 \(\int \frac{x^3}{1+2 x+x^2} \, dx\)

Optimal. Leaf size=22 \[ \frac{x^2}{2}-2 x+\frac{1}{x+1}+3 \log (x+1) \]

[Out]

-2*x + x^2/2 + (1 + x)^(-1) + 3*Log[1 + x]

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Rubi [A]  time = 0.0293626, antiderivative size = 22, 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 \[ \frac{x^2}{2}-2 x+\frac{1}{x+1}+3 \log (x+1) \]

Antiderivative was successfully verified.

[In]  Int[x^3/(1 + 2*x + x^2),x]

[Out]

-2*x + x^2/2 + (1 + x)^(-1) + 3*Log[1 + x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ 3 \log{\left (x + 1 \right )} + 4 \int \left (- \frac{1}{2}\right )\, dx + \int x\, dx + \frac{1}{x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(x**2+2*x+1),x)

[Out]

3*log(x + 1) + 4*Integral(-1/2, x) + Integral(x, x) + 1/(x + 1)

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Mathematica [A]  time = 0.0172247, size = 26, normalized size = 1.18 \[ \frac{1}{2} (x+1)^2-3 (x+1)+\frac{1}{x+1}+3 \log (x+1) \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(1 + 2*x + x^2),x]

[Out]

(1 + x)^(-1) - 3*(1 + x) + (1 + x)^2/2 + 3*Log[1 + x]

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Maple [A]  time = 0.009, size = 21, normalized size = 1. \[ -2\,x+{\frac{{x}^{2}}{2}}+ \left ( 1+x \right ) ^{-1}+3\,\ln \left ( 1+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(x^2+2*x+1),x)

[Out]

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

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Maxima [A]  time = 0.680975, size = 27, normalized size = 1.23 \[ \frac{1}{2} \, x^{2} - 2 \, x + \frac{1}{x + 1} + 3 \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.199815, size = 39, normalized size = 1.77 \[ \frac{x^{3} - 3 \, x^{2} + 6 \,{\left (x + 1\right )} \log \left (x + 1\right ) - 4 \, x + 2}{2 \,{\left (x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.140783, size = 19, normalized size = 0.86 \[ \frac{x^{2}}{2} - 2 x + 3 \log{\left (x + 1 \right )} + \frac{1}{x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(x**2+2*x+1),x)

[Out]

x**2/2 - 2*x + 3*log(x + 1) + 1/(x + 1)

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GIAC/XCAS [A]  time = 0.203993, size = 28, normalized size = 1.27 \[ \frac{1}{2} \, x^{2} - 2 \, x + \frac{1}{x + 1} + 3 \,{\rm ln}\left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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