3.1270 \(\int \frac{1}{x \left (a+b x^5\right )} \, dx\)

Optimal. Leaf size=22 \[ \frac{\log (x)}{a}-\frac{\log \left (a+b x^5\right )}{5 a} \]

[Out]

Log[x]/a - Log[a + b*x^5]/(5*a)

_______________________________________________________________________________________

Rubi [A]  time = 0.0323554, antiderivative size = 22, 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 \[ \frac{\log (x)}{a}-\frac{\log \left (a+b x^5\right )}{5 a} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a + b*x^5)),x]

[Out]

Log[x]/a - Log[a + b*x^5]/(5*a)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.43705, size = 19, normalized size = 0.86 \[ \frac{\log{\left (x^{5} \right )}}{5 a} - \frac{\log{\left (a + b x^{5} \right )}}{5 a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.00885009, size = 22, normalized size = 1. \[ \frac{\log (x)}{a}-\frac{\log \left (a+b x^5\right )}{5 a} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(a + b*x^5)),x]

[Out]

Log[x]/a - Log[a + b*x^5]/(5*a)

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 21, normalized size = 1. \[{\frac{\ln \left ( x \right ) }{a}}-{\frac{\ln \left ( b{x}^{5}+a \right ) }{5\,a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(b*x^5+a),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.42725, size = 31, normalized size = 1.41 \[ -\frac{\log \left (b x^{5} + a\right )}{5 \, a} + \frac{\log \left (x^{5}\right )}{5 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.218365, size = 24, normalized size = 1.09 \[ -\frac{\log \left (b x^{5} + a\right ) - 5 \, \log \left (x\right )}{5 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5*(log(b*x^5 + a) - 5*log(x))/a

_______________________________________________________________________________________

Sympy [A]  time = 0.787942, size = 15, normalized size = 0.68 \[ \frac{\log{\left (x \right )}}{a} - \frac{\log{\left (\frac{a}{b} + x^{5} \right )}}{5 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(b*x**5+a),x)

[Out]

log(x)/a - log(a/b + x**5)/(5*a)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.22134, size = 30, normalized size = 1.36 \[ -\frac{{\rm ln}\left ({\left | b x^{5} + a \right |}\right )}{5 \, a} + \frac{{\rm ln}\left ({\left | x \right |}\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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