3.151 \(\int \frac{x^{11}}{3+4 x^3+x^6} \, dx\)

Optimal. Leaf size=35 \[ \frac{x^6}{6}-\frac{4 x^3}{3}-\frac{1}{6} \log \left (x^3+1\right )+\frac{9}{2} \log \left (x^3+3\right ) \]

[Out]

(-4*x^3)/3 + x^6/6 - Log[1 + x^3]/6 + (9*Log[3 + x^3])/2

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Rubi [A]  time = 0.0561295, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{x^6}{6}-\frac{4 x^3}{3}-\frac{1}{6} \log \left (x^3+1\right )+\frac{9}{2} \log \left (x^3+3\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^11/(3 + 4*x^3 + x^6),x]

[Out]

(-4*x^3)/3 + x^6/6 - Log[1 + x^3]/6 + (9*Log[3 + x^3])/2

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{4 x^{3}}{3} - \frac{\log{\left (x^{3} + 1 \right )}}{6} + \frac{9 \log{\left (x^{3} + 3 \right )}}{2} + \frac{\int ^{x^{3}} x\, dx}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**11/(x**6+4*x**3+3),x)

[Out]

-4*x**3/3 - log(x**3 + 1)/6 + 9*log(x**3 + 3)/2 + Integral(x, (x, x**3))/3

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Mathematica [A]  time = 0.010818, size = 35, normalized size = 1. \[ \frac{x^6}{6}-\frac{4 x^3}{3}-\frac{1}{6} \log \left (x^3+1\right )+\frac{9}{2} \log \left (x^3+3\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^11/(3 + 4*x^3 + x^6),x]

[Out]

(-4*x^3)/3 + x^6/6 - Log[1 + x^3]/6 + (9*Log[3 + x^3])/2

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Maple [A]  time = 0.009, size = 28, normalized size = 0.8 \[ -{\frac{4\,{x}^{3}}{3}}+{\frac{{x}^{6}}{6}}-{\frac{\ln \left ({x}^{3}+1 \right ) }{6}}+{\frac{9\,\ln \left ({x}^{3}+3 \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11/(x^6+4*x^3+3),x)

[Out]

-4/3*x^3+1/6*x^6-1/6*ln(x^3+1)+9/2*ln(x^3+3)

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Maxima [A]  time = 0.78124, size = 36, normalized size = 1.03 \[ \frac{1}{6} \, x^{6} - \frac{4}{3} \, x^{3} + \frac{9}{2} \, \log \left (x^{3} + 3\right ) - \frac{1}{6} \, \log \left (x^{3} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(x^6 + 4*x^3 + 3),x, algorithm="maxima")

[Out]

1/6*x^6 - 4/3*x^3 + 9/2*log(x^3 + 3) - 1/6*log(x^3 + 1)

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Fricas [A]  time = 0.261441, size = 36, normalized size = 1.03 \[ \frac{1}{6} \, x^{6} - \frac{4}{3} \, x^{3} + \frac{9}{2} \, \log \left (x^{3} + 3\right ) - \frac{1}{6} \, \log \left (x^{3} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(x^6 + 4*x^3 + 3),x, algorithm="fricas")

[Out]

1/6*x^6 - 4/3*x^3 + 9/2*log(x^3 + 3) - 1/6*log(x^3 + 1)

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Sympy [A]  time = 0.256044, size = 29, normalized size = 0.83 \[ \frac{x^{6}}{6} - \frac{4 x^{3}}{3} - \frac{\log{\left (x^{3} + 1 \right )}}{6} + \frac{9 \log{\left (x^{3} + 3 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**11/(x**6+4*x**3+3),x)

[Out]

x**6/6 - 4*x**3/3 - log(x**3 + 1)/6 + 9*log(x**3 + 3)/2

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GIAC/XCAS [A]  time = 0.285126, size = 39, normalized size = 1.11 \[ \frac{1}{6} \, x^{6} - \frac{4}{3} \, x^{3} + \frac{9}{2} \,{\rm ln}\left ({\left | x^{3} + 3 \right |}\right ) - \frac{1}{6} \,{\rm ln}\left ({\left | x^{3} + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(x^6 + 4*x^3 + 3),x, algorithm="giac")

[Out]

1/6*x^6 - 4/3*x^3 + 9/2*ln(abs(x^3 + 3)) - 1/6*ln(abs(x^3 + 1))