3.1878 \(\int \frac{1}{\left (a+\frac{b}{x^2}\right )^3 x} \, dx\)

Optimal. Leaf size=49 \[ -\frac{b^2}{4 a^3 \left (a x^2+b\right )^2}+\frac{b}{a^3 \left (a x^2+b\right )}+\frac{\log \left (a x^2+b\right )}{2 a^3} \]

[Out]

-b^2/(4*a^3*(b + a*x^2)^2) + b/(a^3*(b + a*x^2)) + Log[b + a*x^2]/(2*a^3)

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Rubi [A]  time = 0.101023, antiderivative size = 49, 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 \[ -\frac{b^2}{4 a^3 \left (a x^2+b\right )^2}+\frac{b}{a^3 \left (a x^2+b\right )}+\frac{\log \left (a x^2+b\right )}{2 a^3} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^2)^3*x),x]

[Out]

-b^2/(4*a^3*(b + a*x^2)^2) + b/(a^3*(b + a*x^2)) + Log[b + a*x^2]/(2*a^3)

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Rubi in Sympy [A]  time = 13.211, size = 41, normalized size = 0.84 \[ - \frac{b^{2}}{4 a^{3} \left (a x^{2} + b\right )^{2}} + \frac{b}{a^{3} \left (a x^{2} + b\right )} + \frac{\log{\left (a x^{2} + b \right )}}{2 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**2)**3/x,x)

[Out]

-b**2/(4*a**3*(a*x**2 + b)**2) + b/(a**3*(a*x**2 + b)) + log(a*x**2 + b)/(2*a**3
)

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Mathematica [A]  time = 0.0275467, size = 39, normalized size = 0.8 \[ \frac{\frac{b \left (4 a x^2+3 b\right )}{\left (a x^2+b\right )^2}+2 \log \left (a x^2+b\right )}{4 a^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x^2)^3*x),x]

[Out]

((b*(3*b + 4*a*x^2))/(b + a*x^2)^2 + 2*Log[b + a*x^2])/(4*a^3)

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Maple [A]  time = 0.012, size = 46, normalized size = 0.9 \[ -{\frac{{b}^{2}}{4\,{a}^{3} \left ( a{x}^{2}+b \right ) ^{2}}}+{\frac{b}{{a}^{3} \left ( a{x}^{2}+b \right ) }}+{\frac{\ln \left ( a{x}^{2}+b \right ) }{2\,{a}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)^3/x,x)

[Out]

-1/4*b^2/a^3/(a*x^2+b)^2+b/a^3/(a*x^2+b)+1/2*ln(a*x^2+b)/a^3

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Maxima [A]  time = 1.42738, size = 74, normalized size = 1.51 \[ \frac{4 \, a b x^{2} + 3 \, b^{2}}{4 \,{\left (a^{5} x^{4} + 2 \, a^{4} b x^{2} + a^{3} b^{2}\right )}} + \frac{\log \left (a x^{2} + b\right )}{2 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^3*x),x, algorithm="maxima")

[Out]

1/4*(4*a*b*x^2 + 3*b^2)/(a^5*x^4 + 2*a^4*b*x^2 + a^3*b^2) + 1/2*log(a*x^2 + b)/a
^3

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Fricas [A]  time = 0.226576, size = 93, normalized size = 1.9 \[ \frac{4 \, a b x^{2} + 3 \, b^{2} + 2 \,{\left (a^{2} x^{4} + 2 \, a b x^{2} + b^{2}\right )} \log \left (a x^{2} + b\right )}{4 \,{\left (a^{5} x^{4} + 2 \, a^{4} b x^{2} + a^{3} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^3*x),x, algorithm="fricas")

[Out]

1/4*(4*a*b*x^2 + 3*b^2 + 2*(a^2*x^4 + 2*a*b*x^2 + b^2)*log(a*x^2 + b))/(a^5*x^4
+ 2*a^4*b*x^2 + a^3*b^2)

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Sympy [A]  time = 1.89388, size = 53, normalized size = 1.08 \[ \frac{4 a b x^{2} + 3 b^{2}}{4 a^{5} x^{4} + 8 a^{4} b x^{2} + 4 a^{3} b^{2}} + \frac{\log{\left (a x^{2} + b \right )}}{2 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**2)**3/x,x)

[Out]

(4*a*b*x**2 + 3*b**2)/(4*a**5*x**4 + 8*a**4*b*x**2 + 4*a**3*b**2) + log(a*x**2 +
b)/(2*a**3)

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GIAC/XCAS [A]  time = 0.223375, size = 57, normalized size = 1.16 \[ \frac{{\rm ln}\left ({\left | a x^{2} + b \right |}\right )}{2 \, a^{3}} - \frac{3 \, a x^{4} + 2 \, b x^{2}}{4 \,{\left (a x^{2} + b\right )}^{2} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^3*x),x, algorithm="giac")

[Out]

1/2*ln(abs(a*x^2 + b))/a^3 - 1/4*(3*a*x^4 + 2*b*x^2)/((a*x^2 + b)^2*a^2)