3.1287 \(\int \frac{1}{x^6 \left (2 b+b x^5\right )} \, dx\)

Optimal. Leaf size=33 \[ -\frac{1}{10 b x^5}+\frac{\log \left (x^5+2\right )}{20 b}-\frac{\log (x)}{4 b} \]

[Out]

-1/(10*b*x^5) - Log[x]/(4*b) + Log[2 + x^5]/(20*b)

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Rubi [A]  time = 0.0451624, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{1}{10 b x^5}+\frac{\log \left (x^5+2\right )}{20 b}-\frac{\log (x)}{4 b} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^6*(2*b + b*x^5)),x]

[Out]

-1/(10*b*x^5) - Log[x]/(4*b) + Log[2 + x^5]/(20*b)

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Rubi in Sympy [A]  time = 8.06092, size = 26, normalized size = 0.79 \[ - \frac{\log{\left (x^{5} \right )}}{20 b} + \frac{\log{\left (x^{5} + 2 \right )}}{20 b} - \frac{1}{10 b x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**6/(b*x**5+2*b),x)

[Out]

-log(x**5)/(20*b) + log(x**5 + 2)/(20*b) - 1/(10*b*x**5)

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Mathematica [A]  time = 0.00768055, size = 33, normalized size = 1. \[ -\frac{1}{10 b x^5}+\frac{\log \left (x^5+2\right )}{20 b}-\frac{\log (x)}{4 b} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^6*(2*b + b*x^5)),x]

[Out]

-1/(10*b*x^5) - Log[x]/(4*b) + Log[2 + x^5]/(20*b)

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Maple [A]  time = 0.01, size = 28, normalized size = 0.9 \[ -{\frac{1}{10\,b{x}^{5}}}-{\frac{\ln \left ( x \right ) }{4\,b}}+{\frac{\ln \left ({x}^{5}+2 \right ) }{20\,b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^6/(b*x^5+2*b),x)

[Out]

-1/10/b/x^5-1/4*ln(x)/b+1/20*ln(x^5+2)/b

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Maxima [A]  time = 1.44511, size = 39, normalized size = 1.18 \[ \frac{\log \left (x^{5} + 2\right )}{20 \, b} - \frac{\log \left (x^{5}\right )}{20 \, b} - \frac{1}{10 \, b x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^5 + 2*b)*x^6),x, algorithm="maxima")

[Out]

1/20*log(x^5 + 2)/b - 1/20*log(x^5)/b - 1/10/(b*x^5)

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Fricas [A]  time = 0.220432, size = 36, normalized size = 1.09 \[ \frac{x^{5} \log \left (x^{5} + 2\right ) - 5 \, x^{5} \log \left (x\right ) - 2}{20 \, b x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^5 + 2*b)*x^6),x, algorithm="fricas")

[Out]

1/20*(x^5*log(x^5 + 2) - 5*x^5*log(x) - 2)/(b*x^5)

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Sympy [A]  time = 2.80609, size = 24, normalized size = 0.73 \[ - \frac{\log{\left (x \right )}}{4 b} + \frac{\log{\left (x^{5} + 2 \right )}}{20 b} - \frac{1}{10 b x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**6/(b*x**5+2*b),x)

[Out]

-log(x)/(4*b) + log(x**5 + 2)/(20*b) - 1/(10*b*x**5)

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GIAC/XCAS [A]  time = 0.239208, size = 46, normalized size = 1.39 \[ \frac{{\rm ln}\left ({\left | x^{5} + 2 \right |}\right )}{20 \, b} - \frac{{\rm ln}\left ({\left | x \right |}\right )}{4 \, b} + \frac{x^{5} - 2}{20 \, b x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^5 + 2*b)*x^6),x, algorithm="giac")

[Out]

1/20*ln(abs(x^5 + 2))/b - 1/4*ln(abs(x))/b + 1/20*(x^5 - 2)/(b*x^5)