3.1867 \(\int \frac{1}{\left (a+\frac{b}{x^2}\right )^2 x^5} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0757825, antiderivative size = 38, 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 \[ -\frac{\log \left (a x^2+b\right )}{2 b^2}+\frac{1}{2 b \left (a x^2+b\right )}+\frac{\log (x)}{b^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^2)^2*x^5),x]

[Out]

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

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Rubi in Sympy [A]  time = 10.0786, size = 34, normalized size = 0.89 \[ \frac{1}{2 b \left (a x^{2} + b\right )} + \frac{\log{\left (x^{2} \right )}}{2 b^{2}} - \frac{\log{\left (a x^{2} + b \right )}}{2 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.023608, size = 33, normalized size = 0.87 \[ \frac{\frac{b}{a x^2+b}-\log \left (a x^2+b\right )+2 \log (x)}{2 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.015, size = 35, normalized size = 0.9 \[{\frac{1}{2\,b \left ( a{x}^{2}+b \right ) }}+{\frac{\ln \left ( x \right ) }{{b}^{2}}}-{\frac{\ln \left ( a{x}^{2}+b \right ) }{2\,{b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)^2/x^5,x)

[Out]

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

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Maxima [A]  time = 1.43685, size = 50, normalized size = 1.32 \[ \frac{1}{2 \,{\left (a b x^{2} + b^{2}\right )}} - \frac{\log \left (a x^{2} + b\right )}{2 \, b^{2}} + \frac{\log \left (x^{2}\right )}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.240162, size = 63, normalized size = 1.66 \[ -\frac{{\left (a x^{2} + b\right )} \log \left (a x^{2} + b\right ) - 2 \,{\left (a x^{2} + b\right )} \log \left (x\right ) - b}{2 \,{\left (a b^{2} x^{2} + b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.69092, size = 34, normalized size = 0.89 \[ \frac{1}{2 a b x^{2} + 2 b^{2}} + \frac{\log{\left (x \right )}}{b^{2}} - \frac{\log{\left (x^{2} + \frac{b}{a} \right )}}{2 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.228627, size = 63, normalized size = 1.66 \[ \frac{{\rm ln}\left (x^{2}\right )}{2 \, b^{2}} - \frac{{\rm ln}\left ({\left | a x^{2} + b \right |}\right )}{2 \, b^{2}} + \frac{a x^{2} + 2 \, b}{2 \,{\left (a x^{2} + b\right )} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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