3.1863 \(\int \frac{1}{\left (a+\frac{b}{x^2}\right )^2 x} \, dx\)

Optimal. Leaf size=33 \[ \frac{b}{2 a^2 \left (a x^2+b\right )}+\frac{\log \left (a x^2+b\right )}{2 a^2} \]

[Out]

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

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Rubi [A]  time = 0.0681033, antiderivative size = 33, 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 \[ \frac{b}{2 a^2 \left (a x^2+b\right )}+\frac{\log \left (a x^2+b\right )}{2 a^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^2)^2*x),x]

[Out]

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

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Rubi in Sympy [A]  time = 9.30753, size = 26, normalized size = 0.79 \[ \frac{b}{2 a^{2} \left (a x^{2} + b\right )} + \frac{\log{\left (a x^{2} + b \right )}}{2 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0146271, size = 27, normalized size = 0.82 \[ \frac{\frac{b}{a x^2+b}+\log \left (a x^2+b\right )}{2 a^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 30, normalized size = 0.9 \[{\frac{b}{2\,{a}^{2} \left ( a{x}^{2}+b \right ) }}+{\frac{\ln \left ( a{x}^{2}+b \right ) }{2\,{a}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)^2/x,x)

[Out]

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

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Maxima [A]  time = 1.44062, size = 43, normalized size = 1.3 \[ \frac{b}{2 \,{\left (a^{3} x^{2} + a^{2} b\right )}} + \frac{\log \left (a x^{2} + b\right )}{2 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.230099, size = 47, normalized size = 1.42 \[ \frac{{\left (a x^{2} + b\right )} \log \left (a x^{2} + b\right ) + b}{2 \,{\left (a^{3} x^{2} + a^{2} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.36042, size = 29, normalized size = 0.88 \[ \frac{b}{2 a^{3} x^{2} + 2 a^{2} b} + \frac{\log{\left (a x^{2} + b \right )}}{2 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**2)**2/x,x)

[Out]

b/(2*a**3*x**2 + 2*a**2*b) + log(a*x**2 + b)/(2*a**2)

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GIAC/XCAS [A]  time = 0.224437, size = 43, normalized size = 1.3 \[ -\frac{x^{2}}{2 \,{\left (a x^{2} + b\right )} a} + \frac{{\rm ln}\left ({\left | a x^{2} + b \right |}\right )}{2 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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