3.156 \(\int \frac{x^5}{-4+x^2+3 x^4} \, dx\)

Optimal. Leaf size=32 \[ \frac{x^2}{6}+\frac{1}{14} \log \left (1-x^2\right )-\frac{8}{63} \log \left (3 x^2+4\right ) \]

[Out]

x^2/6 + Log[1 - x^2]/14 - (8*Log[4 + 3*x^2])/63

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Rubi [A]  time = 0.0551289, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{x^2}{6}+\frac{1}{14} \log \left (1-x^2\right )-\frac{8}{63} \log \left (3 x^2+4\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^5/(-4 + x^2 + 3*x^4),x]

[Out]

x^2/6 + Log[1 - x^2]/14 - (8*Log[4 + 3*x^2])/63

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Rubi in Sympy [A]  time = 5.69744, size = 24, normalized size = 0.75 \[ \frac{x^{2}}{6} + \frac{\log{\left (- x^{2} + 1 \right )}}{14} - \frac{8 \log{\left (3 x^{2} + 4 \right )}}{63} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(3*x**4+x**2-4),x)

[Out]

x**2/6 + log(-x**2 + 1)/14 - 8*log(3*x**2 + 4)/63

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Mathematica [A]  time = 0.0079071, size = 32, normalized size = 1. \[ \frac{x^2}{6}+\frac{1}{14} \log \left (1-x^2\right )-\frac{8}{63} \log \left (3 x^2+4\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(-4 + x^2 + 3*x^4),x]

[Out]

x^2/6 + Log[1 - x^2]/14 - (8*Log[4 + 3*x^2])/63

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Maple [A]  time = 0.01, size = 25, normalized size = 0.8 \[{\frac{{x}^{2}}{6}}+{\frac{\ln \left ({x}^{2}-1 \right ) }{14}}-{\frac{8\,\ln \left ( 3\,{x}^{2}+4 \right ) }{63}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(3*x^4+x^2-4),x)

[Out]

1/6*x^2+1/14*ln(x^2-1)-8/63*ln(3*x^2+4)

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Maxima [A]  time = 1.35628, size = 32, normalized size = 1. \[ \frac{1}{6} \, x^{2} - \frac{8}{63} \, \log \left (3 \, x^{2} + 4\right ) + \frac{1}{14} \, \log \left (x^{2} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(3*x^4 + x^2 - 4),x, algorithm="maxima")

[Out]

1/6*x^2 - 8/63*log(3*x^2 + 4) + 1/14*log(x^2 - 1)

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Fricas [A]  time = 0.198794, size = 32, normalized size = 1. \[ \frac{1}{6} \, x^{2} - \frac{8}{63} \, \log \left (3 \, x^{2} + 4\right ) + \frac{1}{14} \, \log \left (x^{2} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(3*x^4 + x^2 - 4),x, algorithm="fricas")

[Out]

1/6*x^2 - 8/63*log(3*x^2 + 4) + 1/14*log(x^2 - 1)

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Sympy [A]  time = 0.13361, size = 24, normalized size = 0.75 \[ \frac{x^{2}}{6} + \frac{\log{\left (x^{2} - 1 \right )}}{14} - \frac{8 \log{\left (x^{2} + \frac{4}{3} \right )}}{63} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(3*x**4+x**2-4),x)

[Out]

x**2/6 + log(x**2 - 1)/14 - 8*log(x**2 + 4/3)/63

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GIAC/XCAS [A]  time = 0.201976, size = 34, normalized size = 1.06 \[ \frac{1}{6} \, x^{2} - \frac{8}{63} \,{\rm ln}\left (3 \, x^{2} + 4\right ) + \frac{1}{14} \,{\rm ln}\left ({\left | x^{2} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(3*x^4 + x^2 - 4),x, algorithm="giac")

[Out]

1/6*x^2 - 8/63*ln(3*x^2 + 4) + 1/14*ln(abs(x^2 - 1))