3.1861 \(\int \frac{x}{\left (a+\frac{b}{x^2}\right )^2} \, dx\)

Optimal. Leaf size=44 \[ -\frac{b^2}{2 a^3 \left (a x^2+b\right )}-\frac{b \log \left (a x^2+b\right )}{a^3}+\frac{x^2}{2 a^2} \]

[Out]

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

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Rubi [A]  time = 0.0867551, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{b^2}{2 a^3 \left (a x^2+b\right )}-\frac{b \log \left (a x^2+b\right )}{a^3}+\frac{x^2}{2 a^2} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b/x^2)^2,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\int ^{x^{2}} \frac{1}{a^{2}}\, dx}{2} - \frac{b^{2}}{2 a^{3} \left (a x^{2} + b\right )} - \frac{b \log{\left (a x^{2} + b \right )}}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0289089, size = 38, normalized size = 0.86 \[ \frac{-\frac{b^2}{a x^2+b}-2 b \log \left (a x^2+b\right )+a x^2}{2 a^3} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(a + b/x^2)^2,x]

[Out]

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

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Maple [A]  time = 0.013, size = 41, normalized size = 0.9 \[{\frac{{x}^{2}}{2\,{a}^{2}}}-{\frac{{b}^{2}}{2\,{a}^{3} \left ( a{x}^{2}+b \right ) }}-{\frac{b\ln \left ( a{x}^{2}+b \right ) }{{a}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(a+b/x^2)^2,x)

[Out]

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

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Maxima [A]  time = 1.43775, size = 58, normalized size = 1.32 \[ -\frac{b^{2}}{2 \,{\left (a^{4} x^{2} + a^{3} b\right )}} + \frac{x^{2}}{2 \, a^{2}} - \frac{b \log \left (a x^{2} + b\right )}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.225497, size = 76, normalized size = 1.73 \[ \frac{a^{2} x^{4} + a b x^{2} - b^{2} - 2 \,{\left (a b x^{2} + b^{2}\right )} \log \left (a x^{2} + b\right )}{2 \,{\left (a^{4} x^{2} + a^{3} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.54483, size = 39, normalized size = 0.89 \[ - \frac{b^{2}}{2 a^{4} x^{2} + 2 a^{3} b} + \frac{x^{2}}{2 a^{2}} - \frac{b \log{\left (a x^{2} + b \right )}}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(a+b/x**2)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.223274, size = 55, normalized size = 1.25 \[ \frac{x^{2}}{2 \, a^{2}} - \frac{b{\rm ln}\left ({\left | a x^{2} + b \right |}\right )}{a^{3}} - \frac{b^{2}}{2 \,{\left (a x^{2} + b\right )} a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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