3.1852 \(\int \frac{1}{\left (a+\frac{b}{x^2}\right ) x^5} \, dx\)

Optimal. Leaf size=35 \[ \frac{a \log \left (a x^2+b\right )}{2 b^2}-\frac{a \log (x)}{b^2}-\frac{1}{2 b x^2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0688594, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{a \log \left (a x^2+b\right )}{2 b^2}-\frac{a \log (x)}{b^2}-\frac{1}{2 b x^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^2)*x^5),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 9.68781, size = 34, normalized size = 0.97 \[ - \frac{a \log{\left (x^{2} \right )}}{2 b^{2}} + \frac{a \log{\left (a x^{2} + b \right )}}{2 b^{2}} - \frac{1}{2 b x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0119715, size = 35, normalized size = 1. \[ \frac{a \log \left (a x^2+b\right )}{2 b^2}-\frac{a \log (x)}{b^2}-\frac{1}{2 b x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 32, normalized size = 0.9 \[ -{\frac{1}{2\,b{x}^{2}}}-{\frac{a\ln \left ( x \right ) }{{b}^{2}}}+{\frac{a\ln \left ( a{x}^{2}+b \right ) }{2\,{b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)/x^5,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.54631, size = 45, normalized size = 1.29 \[ \frac{a \log \left (a x^{2} + b\right )}{2 \, b^{2}} - \frac{a \log \left (x^{2}\right )}{2 \, b^{2}} - \frac{1}{2 \, b x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.237591, size = 45, normalized size = 1.29 \[ \frac{a x^{2} \log \left (a x^{2} + b\right ) - 2 \, a x^{2} \log \left (x\right ) - b}{2 \, b^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 1.57859, size = 31, normalized size = 0.89 \[ - \frac{a \log{\left (x \right )}}{b^{2}} + \frac{a \log{\left (x^{2} + \frac{b}{a} \right )}}{2 b^{2}} - \frac{1}{2 b x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**2)/x**5,x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.225572, size = 58, normalized size = 1.66 \[ -\frac{a{\rm ln}\left (x^{2}\right )}{2 \, b^{2}} + \frac{a{\rm ln}\left ({\left | a x^{2} + b \right |}\right )}{2 \, b^{2}} + \frac{a x^{2} - b}{2 \, b^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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