3.1280 \(\int \frac{1}{x \left (a+b x^5\right )^2} \, dx\)

Optimal. Leaf size=38 \[ -\frac{\log \left (a+b x^5\right )}{5 a^2}+\frac{\log (x)}{a^2}+\frac{1}{5 a \left (a+b x^5\right )} \]

[Out]

1/(5*a*(a + b*x^5)) + Log[x]/a^2 - Log[a + b*x^5]/(5*a^2)

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Rubi [A]  time = 0.0613219, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{\log \left (a+b x^5\right )}{5 a^2}+\frac{\log (x)}{a^2}+\frac{1}{5 a \left (a+b x^5\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a + b*x^5)^2),x]

[Out]

1/(5*a*(a + b*x^5)) + Log[x]/a^2 - Log[a + b*x^5]/(5*a^2)

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Rubi in Sympy [A]  time = 8.34999, size = 34, normalized size = 0.89 \[ \frac{1}{5 a \left (a + b x^{5}\right )} + \frac{\log{\left (x^{5} \right )}}{5 a^{2}} - \frac{\log{\left (a + b x^{5} \right )}}{5 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0234349, size = 33, normalized size = 0.87 \[ \frac{\frac{a}{a+b x^5}-\log \left (a+b x^5\right )+5 \log (x)}{5 a^2} \]

Antiderivative was successfully verified.

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

[Out]

(a/(a + b*x^5) + 5*Log[x] - Log[a + b*x^5])/(5*a^2)

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Maple [A]  time = 0.011, size = 35, normalized size = 0.9 \[{\frac{1}{5\,a \left ( b{x}^{5}+a \right ) }}+{\frac{\ln \left ( x \right ) }{{a}^{2}}}-{\frac{\ln \left ( b{x}^{5}+a \right ) }{5\,{a}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(b*x^5+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.43865, size = 50, normalized size = 1.32 \[ \frac{1}{5 \,{\left (a b x^{5} + a^{2}\right )}} - \frac{\log \left (b x^{5} + a\right )}{5 \, a^{2}} + \frac{\log \left (x^{5}\right )}{5 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.218849, size = 63, normalized size = 1.66 \[ -\frac{{\left (b x^{5} + a\right )} \log \left (b x^{5} + a\right ) - 5 \,{\left (b x^{5} + a\right )} \log \left (x\right ) - a}{5 \,{\left (a^{2} b x^{5} + a^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5*((b*x^5 + a)*log(b*x^5 + a) - 5*(b*x^5 + a)*log(x) - a)/(a^2*b*x^5 + a^3)

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Sympy [A]  time = 3.51443, size = 34, normalized size = 0.89 \[ \frac{1}{5 a^{2} + 5 a b x^{5}} + \frac{\log{\left (x \right )}}{a^{2}} - \frac{\log{\left (\frac{a}{b} + x^{5} \right )}}{5 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(b*x**5+a)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.230501, size = 61, normalized size = 1.61 \[ -\frac{{\rm ln}\left ({\left | b x^{5} + a \right |}\right )}{5 \, a^{2}} + \frac{{\rm ln}\left ({\left | x \right |}\right )}{a^{2}} + \frac{b x^{5} + 2 \, a}{5 \,{\left (b x^{5} + a\right )} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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