3.1876 \(\int \frac{x}{\left (a+\frac{b}{x^2}\right )^3} \, dx\)

Optimal. Leaf size=65 \[ \frac{b^3}{4 a^4 \left (a x^2+b\right )^2}-\frac{3 b^2}{2 a^4 \left (a x^2+b\right )}-\frac{3 b \log \left (a x^2+b\right )}{2 a^4}+\frac{x^2}{2 a^3} \]

[Out]

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

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Rubi [A]  time = 0.118351, antiderivative size = 65, 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 \[ \frac{b^3}{4 a^4 \left (a x^2+b\right )^2}-\frac{3 b^2}{2 a^4 \left (a x^2+b\right )}-\frac{3 b \log \left (a x^2+b\right )}{2 a^4}+\frac{x^2}{2 a^3} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b/x^2)^3,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\int ^{x^{2}} \frac{1}{a^{3}}\, dx}{2} + \frac{b^{3}}{4 a^{4} \left (a x^{2} + b\right )^{2}} - \frac{3 b^{2}}{2 a^{4} \left (a x^{2} + b\right )} - \frac{3 b \log{\left (a x^{2} + b \right )}}{2 a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0988092, size = 48, normalized size = 0.74 \[ -\frac{\frac{b^2 \left (6 a x^2+5 b\right )}{\left (a x^2+b\right )^2}+6 b \log \left (a x^2+b\right )-2 a x^2}{4 a^4} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(a + b/x^2)^3,x]

[Out]

-(-2*a*x^2 + (b^2*(5*b + 6*a*x^2))/(b + a*x^2)^2 + 6*b*Log[b + a*x^2])/(4*a^4)

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Maple [A]  time = 0.014, size = 58, normalized size = 0.9 \[{\frac{{x}^{2}}{2\,{a}^{3}}}+{\frac{{b}^{3}}{4\,{a}^{4} \left ( a{x}^{2}+b \right ) ^{2}}}-{\frac{3\,{b}^{2}}{2\,{a}^{4} \left ( a{x}^{2}+b \right ) }}-{\frac{3\,b\ln \left ( a{x}^{2}+b \right ) }{2\,{a}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(a+b/x^2)^3,x)

[Out]

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

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Maxima [A]  time = 1.44359, size = 89, normalized size = 1.37 \[ -\frac{6 \, a b^{2} x^{2} + 5 \, b^{3}}{4 \,{\left (a^{6} x^{4} + 2 \, a^{5} b x^{2} + a^{4} b^{2}\right )}} + \frac{x^{2}}{2 \, a^{3}} - \frac{3 \, b \log \left (a x^{2} + b\right )}{2 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(6*a*b^2*x^2 + 5*b^3)/(a^6*x^4 + 2*a^5*b*x^2 + a^4*b^2) + 1/2*x^2/a^3 - 3/2
*b*log(a*x^2 + b)/a^4

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Fricas [A]  time = 0.221063, size = 123, normalized size = 1.89 \[ \frac{2 \, a^{3} x^{6} + 4 \, a^{2} b x^{4} - 4 \, a b^{2} x^{2} - 5 \, b^{3} - 6 \,{\left (a^{2} b x^{4} + 2 \, a b^{2} x^{2} + b^{3}\right )} \log \left (a x^{2} + b\right )}{4 \,{\left (a^{6} x^{4} + 2 \, a^{5} b x^{2} + a^{4} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(2*a^3*x^6 + 4*a^2*b*x^4 - 4*a*b^2*x^2 - 5*b^3 - 6*(a^2*b*x^4 + 2*a*b^2*x^2
+ b^3)*log(a*x^2 + b))/(a^6*x^4 + 2*a^5*b*x^2 + a^4*b^2)

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Sympy [A]  time = 2.13986, size = 66, normalized size = 1.02 \[ - \frac{6 a b^{2} x^{2} + 5 b^{3}}{4 a^{6} x^{4} + 8 a^{5} b x^{2} + 4 a^{4} b^{2}} + \frac{x^{2}}{2 a^{3}} - \frac{3 b \log{\left (a x^{2} + b \right )}}{2 a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(a+b/x**2)**3,x)

[Out]

-(6*a*b**2*x**2 + 5*b**3)/(4*a**6*x**4 + 8*a**5*b*x**2 + 4*a**4*b**2) + x**2/(2*
a**3) - 3*b*log(a*x**2 + b)/(2*a**4)

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GIAC/XCAS [A]  time = 0.225677, size = 72, normalized size = 1.11 \[ \frac{x^{2}}{2 \, a^{3}} - \frac{3 \, b{\rm ln}\left ({\left | a x^{2} + b \right |}\right )}{2 \, a^{4}} - \frac{6 \, a b^{2} x^{2} + 5 \, b^{3}}{4 \,{\left (a x^{2} + b\right )}^{2} a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x^2/a^3 - 3/2*b*ln(abs(a*x^2 + b))/a^4 - 1/4*(6*a*b^2*x^2 + 5*b^3)/((a*x^2 +
b)^2*a^4)