3.153 \(\int \frac{x^5}{3+4 x^3+x^6} \, dx\)

Optimal. Leaf size=21 \[ \frac{1}{2} \log \left (x^3+3\right )-\frac{1}{6} \log \left (x^3+1\right ) \]

[Out]

-Log[1 + x^3]/6 + Log[3 + x^3]/2

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Rubi [A]  time = 0.0360109, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{1}{2} \log \left (x^3+3\right )-\frac{1}{6} \log \left (x^3+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^5/(3 + 4*x^3 + x^6),x]

[Out]

-Log[1 + x^3]/6 + Log[3 + x^3]/2

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Rubi in Sympy [A]  time = 7.919, size = 15, normalized size = 0.71 \[ - \frac{\log{\left (x^{3} + 1 \right )}}{6} + \frac{\log{\left (x^{3} + 3 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(x**6+4*x**3+3),x)

[Out]

-log(x**3 + 1)/6 + log(x**3 + 3)/2

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Mathematica [A]  time = 0.00720602, size = 21, normalized size = 1. \[ \frac{1}{2} \log \left (x^3+3\right )-\frac{1}{6} \log \left (x^3+1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(3 + 4*x^3 + x^6),x]

[Out]

-Log[1 + x^3]/6 + Log[3 + x^3]/2

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Maple [A]  time = 0.007, size = 18, normalized size = 0.9 \[ -{\frac{\ln \left ({x}^{3}+1 \right ) }{6}}+{\frac{\ln \left ({x}^{3}+3 \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(x^6+4*x^3+3),x)

[Out]

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

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Maxima [A]  time = 0.795143, size = 23, normalized size = 1.1 \[ \frac{1}{2} \, \log \left (x^{3} + 3\right ) - \frac{1}{6} \, \log \left (x^{3} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(x^6 + 4*x^3 + 3),x, algorithm="maxima")

[Out]

1/2*log(x^3 + 3) - 1/6*log(x^3 + 1)

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Fricas [A]  time = 0.25847, size = 23, normalized size = 1.1 \[ \frac{1}{2} \, \log \left (x^{3} + 3\right ) - \frac{1}{6} \, \log \left (x^{3} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(x^6 + 4*x^3 + 3),x, algorithm="fricas")

[Out]

1/2*log(x^3 + 3) - 1/6*log(x^3 + 1)

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Sympy [A]  time = 0.23647, size = 15, normalized size = 0.71 \[ - \frac{\log{\left (x^{3} + 1 \right )}}{6} + \frac{\log{\left (x^{3} + 3 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(x**6+4*x**3+3),x)

[Out]

-log(x**3 + 1)/6 + log(x**3 + 3)/2

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GIAC/XCAS [A]  time = 0.300991, size = 26, normalized size = 1.24 \[ \frac{1}{2} \,{\rm ln}\left ({\left | x^{3} + 3 \right |}\right ) - \frac{1}{6} \,{\rm ln}\left ({\left | x^{3} + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(x^6 + 4*x^3 + 3),x, algorithm="giac")

[Out]

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