3.714 \(\int \frac{x^3}{2 (a+b)+x^4} \, dx\)

Optimal. Leaf size=14 \[ \frac{1}{4} \log \left (2 (a+b)+x^4\right ) \]

[Out]

Log[2*(a + b) + x^4]/4

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Rubi [A]  time = 0.00716794, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{1}{4} \log \left (2 (a+b)+x^4\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^3/(2*(a + b) + x^4),x]

[Out]

Log[2*(a + b) + x^4]/4

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Rubi in Sympy [A]  time = 2.52253, size = 12, normalized size = 0.86 \[ \frac{\log{\left (2 a + 2 b + x^{4} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(x**4+2*a+2*b),x)

[Out]

log(2*a + 2*b + x**4)/4

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Mathematica [A]  time = 0.00625663, size = 15, normalized size = 1.07 \[ \frac{1}{4} \log \left (2 a+2 b+x^4\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(2*(a + b) + x^4),x]

[Out]

Log[2*a + 2*b + x^4]/4

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Maple [A]  time = 0., size = 14, normalized size = 1. \[{\frac{\ln \left ({x}^{4}+2\,a+2\,b \right ) }{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(x^4+2*a+2*b),x)

[Out]

1/4*ln(x^4+2*a+2*b)

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Maxima [A]  time = 1.47282, size = 18, normalized size = 1.29 \[ \frac{1}{4} \, \log \left (x^{4} + 2 \, a + 2 \, b\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*log(x^4 + 2*a + 2*b)

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Fricas [A]  time = 0.220404, size = 18, normalized size = 1.29 \[ \frac{1}{4} \, \log \left (x^{4} + 2 \, a + 2 \, b\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*log(x^4 + 2*a + 2*b)

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Sympy [A]  time = 0.349603, size = 12, normalized size = 0.86 \[ \frac{\log{\left (2 a + 2 b + x^{4} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(x**4+2*a+2*b),x)

[Out]

log(2*a + 2*b + x**4)/4

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GIAC/XCAS [A]  time = 0.225268, size = 19, normalized size = 1.36 \[ \frac{1}{4} \,{\rm ln}\left ({\left | x^{4} + 2 \, a + 2 \, b \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*ln(abs(x^4 + 2*a + 2*b))