3.1871 \(\int \frac{1}{\left (a+\frac{b}{x^2}\right )^2 x^9} \, dx\)

Optimal. Leaf size=66 \[ -\frac{3 a^2 \log \left (a x^2+b\right )}{2 b^4}+\frac{3 a^2 \log (x)}{b^4}+\frac{a^2}{2 b^3 \left (a x^2+b\right )}+\frac{a}{b^3 x^2}-\frac{1}{4 b^2 x^4} \]

[Out]

-1/(4*b^2*x^4) + a/(b^3*x^2) + a^2/(2*b^3*(b + a*x^2)) + (3*a^2*Log[x])/b^4 - (3
*a^2*Log[b + a*x^2])/(2*b^4)

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Rubi [A]  time = 0.118633, antiderivative size = 66, 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 \[ -\frac{3 a^2 \log \left (a x^2+b\right )}{2 b^4}+\frac{3 a^2 \log (x)}{b^4}+\frac{a^2}{2 b^3 \left (a x^2+b\right )}+\frac{a}{b^3 x^2}-\frac{1}{4 b^2 x^4} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^2)^2*x^9),x]

[Out]

-1/(4*b^2*x^4) + a/(b^3*x^2) + a^2/(2*b^3*(b + a*x^2)) + (3*a^2*Log[x])/b^4 - (3
*a^2*Log[b + a*x^2])/(2*b^4)

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Rubi in Sympy [A]  time = 16.0315, size = 66, normalized size = 1. \[ \frac{a^{2}}{2 b^{3} \left (a x^{2} + b\right )} + \frac{3 a^{2} \log{\left (x^{2} \right )}}{2 b^{4}} - \frac{3 a^{2} \log{\left (a x^{2} + b \right )}}{2 b^{4}} + \frac{a}{b^{3} x^{2}} - \frac{1}{4 b^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**2)**2/x**9,x)

[Out]

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

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Mathematica [A]  time = 0.0959095, size = 57, normalized size = 0.86 \[ \frac{-6 a^2 \log \left (a x^2+b\right )+b \left (\frac{2 a^2}{a x^2+b}+\frac{4 a}{x^2}-\frac{b}{x^4}\right )+12 a^2 \log (x)}{4 b^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x^2)^2*x^9),x]

[Out]

(b*(-(b/x^4) + (4*a)/x^2 + (2*a^2)/(b + a*x^2)) + 12*a^2*Log[x] - 6*a^2*Log[b +
a*x^2])/(4*b^4)

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Maple [A]  time = 0.019, size = 61, normalized size = 0.9 \[ -{\frac{1}{4\,{b}^{2}{x}^{4}}}+{\frac{a}{{b}^{3}{x}^{2}}}+{\frac{{a}^{2}}{2\,{b}^{3} \left ( a{x}^{2}+b \right ) }}+3\,{\frac{{a}^{2}\ln \left ( x \right ) }{{b}^{4}}}-{\frac{3\,{a}^{2}\ln \left ( a{x}^{2}+b \right ) }{2\,{b}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)^2/x^9,x)

[Out]

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

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Maxima [A]  time = 1.43896, size = 95, normalized size = 1.44 \[ \frac{6 \, a^{2} x^{4} + 3 \, a b x^{2} - b^{2}}{4 \,{\left (a b^{3} x^{6} + b^{4} x^{4}\right )}} - \frac{3 \, a^{2} \log \left (a x^{2} + b\right )}{2 \, b^{4}} + \frac{3 \, a^{2} \log \left (x^{2}\right )}{2 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^2*x^9),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.23241, size = 122, normalized size = 1.85 \[ \frac{6 \, a^{2} b x^{4} + 3 \, a b^{2} x^{2} - b^{3} - 6 \,{\left (a^{3} x^{6} + a^{2} b x^{4}\right )} \log \left (a x^{2} + b\right ) + 12 \,{\left (a^{3} x^{6} + a^{2} b x^{4}\right )} \log \left (x\right )}{4 \,{\left (a b^{4} x^{6} + b^{5} x^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^2*x^9),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.6295, size = 68, normalized size = 1.03 \[ \frac{3 a^{2} \log{\left (x \right )}}{b^{4}} - \frac{3 a^{2} \log{\left (x^{2} + \frac{b}{a} \right )}}{2 b^{4}} + \frac{6 a^{2} x^{4} + 3 a b x^{2} - b^{2}}{4 a b^{3} x^{6} + 4 b^{4} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**2)**2/x**9,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.235072, size = 116, normalized size = 1.76 \[ \frac{3 \, a^{2}{\rm ln}\left (x^{2}\right )}{2 \, b^{4}} - \frac{3 \, a^{2}{\rm ln}\left ({\left | a x^{2} + b \right |}\right )}{2 \, b^{4}} + \frac{3 \, a^{3} x^{2} + 4 \, a^{2} b}{2 \,{\left (a x^{2} + b\right )} b^{4}} - \frac{9 \, a^{2} x^{4} - 4 \, a b x^{2} + b^{2}}{4 \, b^{4} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^2*x^9),x, algorithm="giac")

[Out]

3/2*a^2*ln(x^2)/b^4 - 3/2*a^2*ln(abs(a*x^2 + b))/b^4 + 1/2*(3*a^3*x^2 + 4*a^2*b)
/((a*x^2 + b)*b^4) - 1/4*(9*a^2*x^4 - 4*a*b*x^2 + b^2)/(b^4*x^4)