3.1286 \(\int \frac{1}{x \left (2 b+b x^5\right )} \, dx\)

Optimal. Leaf size=23 \[ \frac{\log (x)}{2 b}-\frac{\log \left (x^5+2\right )}{10 b} \]

[Out]

Log[x]/(2*b) - Log[2 + x^5]/(10*b)

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Rubi [A]  time = 0.0296823, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{\log (x)}{2 b}-\frac{\log \left (x^5+2\right )}{10 b} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(2*b + b*x^5)),x]

[Out]

Log[x]/(2*b) - Log[2 + x^5]/(10*b)

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Rubi in Sympy [A]  time = 5.87634, size = 17, normalized size = 0.74 \[ \frac{\log{\left (x^{5} \right )}}{10 b} - \frac{\log{\left (x^{5} + 2 \right )}}{10 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(b*x**5+2*b),x)

[Out]

log(x**5)/(10*b) - log(x**5 + 2)/(10*b)

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Mathematica [A]  time = 0.0104145, size = 23, normalized size = 1. \[ \frac{\log (x)}{2 b}-\frac{\log \left (x^5+2\right )}{10 b} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(2*b + b*x^5)),x]

[Out]

Log[x]/(2*b) - Log[2 + x^5]/(10*b)

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Maple [A]  time = 0.006, size = 20, normalized size = 0.9 \[{\frac{\ln \left ( x \right ) }{2\,b}}-{\frac{\ln \left ({x}^{5}+2 \right ) }{10\,b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(b*x^5+2*b),x)

[Out]

1/2*ln(x)/b-1/10*ln(x^5+2)/b

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Maxima [A]  time = 1.44085, size = 28, normalized size = 1.22 \[ -\frac{\log \left (x^{5} + 2\right )}{10 \, b} + \frac{\log \left (x^{5}\right )}{10 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^5 + 2*b)*x),x, algorithm="maxima")

[Out]

-1/10*log(x^5 + 2)/b + 1/10*log(x^5)/b

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Fricas [A]  time = 0.220929, size = 22, normalized size = 0.96 \[ -\frac{\log \left (x^{5} + 2\right ) - 5 \, \log \left (x\right )}{10 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^5 + 2*b)*x),x, algorithm="fricas")

[Out]

-1/10*(log(x^5 + 2) - 5*log(x))/b

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Sympy [A]  time = 0.772315, size = 15, normalized size = 0.65 \[ \frac{\log{\left (x \right )}}{2 b} - \frac{\log{\left (x^{5} + 2 \right )}}{10 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(b*x**5+2*b),x)

[Out]

log(x)/(2*b) - log(x**5 + 2)/(10*b)

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GIAC/XCAS [A]  time = 0.237359, size = 28, normalized size = 1.22 \[ -\frac{{\rm ln}\left ({\left | x^{5} + 2 \right |}\right )}{10 \, b} + \frac{{\rm ln}\left ({\left | x \right |}\right )}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^5 + 2*b)*x),x, algorithm="giac")

[Out]

-1/10*ln(abs(x^5 + 2))/b + 1/2*ln(abs(x))/b