3.1291 \(\int \frac{1}{x \left (3+b x^5\right )} \, dx\)

Optimal. Leaf size=19 \[ \frac{\log (x)}{3}-\frac{1}{15} \log \left (b x^5+3\right ) \]

[Out]

Log[x]/3 - Log[3 + b*x^5]/15

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Rubi [A]  time = 0.0287537, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{\log (x)}{3}-\frac{1}{15} \log \left (b x^5+3\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(3 + b*x^5)),x]

[Out]

Log[x]/3 - Log[3 + b*x^5]/15

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Rubi in Sympy [A]  time = 4.43595, size = 15, normalized size = 0.79 \[ \frac{\log{\left (x^{5} \right )}}{15} - \frac{\log{\left (b x^{5} + 3 \right )}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(b*x**5+3),x)

[Out]

log(x**5)/15 - log(b*x**5 + 3)/15

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Mathematica [A]  time = 0.00661949, size = 19, normalized size = 1. \[ \frac{\log (x)}{3}-\frac{1}{15} \log \left (b x^5+3\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(3 + b*x^5)),x]

[Out]

Log[x]/3 - Log[3 + b*x^5]/15

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Maple [A]  time = 0.006, size = 16, normalized size = 0.8 \[{\frac{\ln \left ( x \right ) }{3}}-{\frac{\ln \left ( b{x}^{5}+3 \right ) }{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(b*x^5+3),x)

[Out]

1/3*ln(x)-1/15*ln(b*x^5+3)

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Maxima [A]  time = 1.44413, size = 23, normalized size = 1.21 \[ -\frac{1}{15} \, \log \left (b x^{5} + 3\right ) + \frac{1}{15} \, \log \left (x^{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^5 + 3)*x),x, algorithm="maxima")

[Out]

-1/15*log(b*x^5 + 3) + 1/15*log(x^5)

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Fricas [A]  time = 0.21541, size = 20, normalized size = 1.05 \[ -\frac{1}{15} \, \log \left (b x^{5} + 3\right ) + \frac{1}{3} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^5 + 3)*x),x, algorithm="fricas")

[Out]

-1/15*log(b*x^5 + 3) + 1/3*log(x)

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Sympy [A]  time = 0.395792, size = 14, normalized size = 0.74 \[ \frac{\log{\left (x \right )}}{3} - \frac{\log{\left (x^{5} + \frac{3}{b} \right )}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(b*x**5+3),x)

[Out]

log(x)/3 - log(x**5 + 3/b)/15

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GIAC/XCAS [A]  time = 0.28261, size = 23, normalized size = 1.21 \[ -\frac{1}{15} \,{\rm ln}\left ({\left | b x^{5} + 3 \right |}\right ) + \frac{1}{3} \,{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^5 + 3)*x),x, algorithm="giac")

[Out]

-1/15*ln(abs(b*x^5 + 3)) + 1/3*ln(abs(x))