3.543 \(\int \frac{243-162 x+108 x^2-72 x^3+48 x^4-32 x^5}{729-64 x^6} \, dx\)

Optimal. Leaf size=10 \[ \frac{1}{2} \log (2 x+3) \]

[Out]

Log[3 + 2*x]/2

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Rubi [A]  time = 0.0162471, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057 \[ \frac{1}{2} \log (2 x+3) \]

Antiderivative was successfully verified.

[In]  Int[(243 - 162*x + 108*x^2 - 72*x^3 + 48*x^4 - 32*x^5)/(729 - 64*x^6),x]

[Out]

Log[3 + 2*x]/2

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Rubi in Sympy [A]  time = 8.25591, size = 7, normalized size = 0.7 \[ \frac{\log{\left (2 x + 3 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-32*x**5+48*x**4-72*x**3+108*x**2-162*x+243)/(-64*x**6+729),x)

[Out]

log(2*x + 3)/2

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Mathematica [A]  time = 0.00180022, size = 10, normalized size = 1. \[ \frac{1}{2} \log (2 x+3) \]

Antiderivative was successfully verified.

[In]  Integrate[(243 - 162*x + 108*x^2 - 72*x^3 + 48*x^4 - 32*x^5)/(729 - 64*x^6),x]

[Out]

Log[3 + 2*x]/2

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Maple [A]  time = 0.002, size = 9, normalized size = 0.9 \[{\frac{\ln \left ( 2\,x+3 \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-32*x^5+48*x^4-72*x^3+108*x^2-162*x+243)/(-64*x^6+729),x)

[Out]

1/2*ln(2*x+3)

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Maxima [A]  time = 1.4215, size = 11, normalized size = 1.1 \[ \frac{1}{2} \, \log \left (2 \, x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((32*x^5 - 48*x^4 + 72*x^3 - 108*x^2 + 162*x - 243)/(64*x^6 - 729),x, algorithm="maxima")

[Out]

1/2*log(2*x + 3)

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Fricas [A]  time = 0.201064, size = 11, normalized size = 1.1 \[ \frac{1}{2} \, \log \left (2 \, x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((32*x^5 - 48*x^4 + 72*x^3 - 108*x^2 + 162*x - 243)/(64*x^6 - 729),x, algorithm="fricas")

[Out]

1/2*log(2*x + 3)

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Sympy [A]  time = 0.058952, size = 7, normalized size = 0.7 \[ \frac{\log{\left (2 x + 3 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-32*x**5+48*x**4-72*x**3+108*x**2-162*x+243)/(-64*x**6+729),x)

[Out]

log(2*x + 3)/2

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GIAC/XCAS [A]  time = 0.216451, size = 12, normalized size = 1.2 \[ \frac{1}{2} \,{\rm ln}\left ({\left | 2 \, x + 3 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((32*x^5 - 48*x^4 + 72*x^3 - 108*x^2 + 162*x - 243)/(64*x^6 - 729),x, algorithm="giac")

[Out]

1/2*ln(abs(2*x + 3))