3.229 \(\int \frac{3 x+3 x^2+x^3}{1+4 x+6 x^2+4 x^3+x^4} \, dx\)

Optimal. Leaf size=14 \[ \frac{1}{3 (x+1)^3}+\log (x+1) \]

[Out]

1/(3*(1 + x)^3) + Log[1 + x]

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Rubi [A]  time = 0.0605462, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{1}{3 (x+1)^3}+\log (x+1) \]

Antiderivative was successfully verified.

[In]  Int[(3*x + 3*x^2 + x^3)/(1 + 4*x + 6*x^2 + 4*x^3 + x^4),x]

[Out]

1/(3*(1 + x)^3) + Log[1 + x]

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Rubi in Sympy [A]  time = 34.3685, size = 12, normalized size = 0.86 \[ \log{\left (x + 1 \right )} + \frac{1}{3 \left (x + 1\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**3+3*x**2+3*x)/(x**4+4*x**3+6*x**2+4*x+1),x)

[Out]

log(x + 1) + 1/(3*(x + 1)**3)

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Mathematica [A]  time = 0.0112096, size = 14, normalized size = 1. \[ \frac{1}{3 (x+1)^3}+\log (x+1) \]

Antiderivative was successfully verified.

[In]  Integrate[(3*x + 3*x^2 + x^3)/(1 + 4*x + 6*x^2 + 4*x^3 + x^4),x]

[Out]

1/(3*(1 + x)^3) + Log[1 + x]

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Maple [A]  time = 0.009, size = 13, normalized size = 0.9 \[{\frac{1}{3\, \left ( 1+x \right ) ^{3}}}+\ln \left ( 1+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^3+3*x^2+3*x)/(x^4+4*x^3+6*x^2+4*x+1),x)

[Out]

1/3/(1+x)^3+ln(1+x)

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Maxima [A]  time = 0.837061, size = 30, normalized size = 2.14 \[ \frac{1}{3 \,{\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )}} + \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 + 3*x^2 + 3*x)/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.26641, size = 51, normalized size = 3.64 \[ \frac{3 \,{\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )} \log \left (x + 1\right ) + 1}{3 \,{\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 + 3*x^2 + 3*x)/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.195147, size = 20, normalized size = 1.43 \[ \log{\left (x + 1 \right )} + \frac{1}{3 x^{3} + 9 x^{2} + 9 x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**3+3*x**2+3*x)/(x**4+4*x**3+6*x**2+4*x+1),x)

[Out]

log(x + 1) + 1/(3*x**3 + 9*x**2 + 9*x + 3)

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GIAC/XCAS [A]  time = 0.259003, size = 18, normalized size = 1.29 \[ \frac{1}{3 \,{\left (x + 1\right )}^{3}} +{\rm ln}\left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 + 3*x^2 + 3*x)/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1),x, algorithm="giac")

[Out]

1/3/(x + 1)^3 + ln(abs(x + 1))