3.1886 \(\int \frac{1}{\left (a+\frac{b}{x^2}\right )^3 x^9} \, dx\)

Optimal. Leaf size=67 \[ \frac{3 a \log \left (a x^2+b\right )}{2 b^4}-\frac{3 a \log (x)}{b^4}-\frac{a}{b^3 \left (a x^2+b\right )}-\frac{a}{4 b^2 \left (a x^2+b\right )^2}-\frac{1}{2 b^3 x^2} \]

[Out]

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

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Rubi [A]  time = 0.126714, antiderivative size = 67, 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 \[ \frac{3 a \log \left (a x^2+b\right )}{2 b^4}-\frac{3 a \log (x)}{b^4}-\frac{a}{b^3 \left (a x^2+b\right )}-\frac{a}{4 b^2 \left (a x^2+b\right )^2}-\frac{1}{2 b^3 x^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^2)^3*x^9),x]

[Out]

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

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Rubi in Sympy [A]  time = 15.3823, size = 66, normalized size = 0.99 \[ - \frac{a}{4 b^{2} \left (a x^{2} + b\right )^{2}} - \frac{a}{b^{3} \left (a x^{2} + b\right )} - \frac{3 a \log{\left (x^{2} \right )}}{2 b^{4}} + \frac{3 a \log{\left (a x^{2} + b \right )}}{2 b^{4}} - \frac{1}{2 b^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**2)**3/x**9,x)

[Out]

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

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Mathematica [A]  time = 0.101068, size = 59, normalized size = 0.88 \[ -\frac{\frac{b \left (6 a^2 x^4+9 a b x^2+2 b^2\right )}{x^2 \left (a x^2+b\right )^2}-6 a \log \left (a x^2+b\right )+12 a \log (x)}{4 b^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x^2)^3*x^9),x]

[Out]

-((b*(2*b^2 + 9*a*b*x^2 + 6*a^2*x^4))/(x^2*(b + a*x^2)^2) + 12*a*Log[x] - 6*a*Lo
g[b + a*x^2])/(4*b^4)

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Maple [A]  time = 0.018, size = 62, normalized size = 0.9 \[ -{\frac{1}{2\,{b}^{3}{x}^{2}}}-{\frac{a}{4\,{b}^{2} \left ( a{x}^{2}+b \right ) ^{2}}}-{\frac{a}{{b}^{3} \left ( a{x}^{2}+b \right ) }}-3\,{\frac{a\ln \left ( x \right ) }{{b}^{4}}}+{\frac{3\,a\ln \left ( a{x}^{2}+b \right ) }{2\,{b}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)^3/x^9,x)

[Out]

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

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Maxima [A]  time = 1.43765, size = 104, normalized size = 1.55 \[ -\frac{6 \, a^{2} x^{4} + 9 \, a b x^{2} + 2 \, b^{2}}{4 \,{\left (a^{2} b^{3} x^{6} + 2 \, a b^{4} x^{4} + b^{5} x^{2}\right )}} + \frac{3 \, a \log \left (a x^{2} + b\right )}{2 \, b^{4}} - \frac{3 \, a \log \left (x^{2}\right )}{2 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^3*x^9),x, algorithm="maxima")

[Out]

-1/4*(6*a^2*x^4 + 9*a*b*x^2 + 2*b^2)/(a^2*b^3*x^6 + 2*a*b^4*x^4 + b^5*x^2) + 3/2
*a*log(a*x^2 + b)/b^4 - 3/2*a*log(x^2)/b^4

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Fricas [A]  time = 0.233763, size = 161, normalized size = 2.4 \[ -\frac{6 \, a^{2} b x^{4} + 9 \, a b^{2} x^{2} + 2 \, b^{3} - 6 \,{\left (a^{3} x^{6} + 2 \, a^{2} b x^{4} + a b^{2} x^{2}\right )} \log \left (a x^{2} + b\right ) + 12 \,{\left (a^{3} x^{6} + 2 \, a^{2} b x^{4} + a b^{2} x^{2}\right )} \log \left (x\right )}{4 \,{\left (a^{2} b^{4} x^{6} + 2 \, a b^{5} x^{4} + b^{6} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^3*x^9),x, algorithm="fricas")

[Out]

-1/4*(6*a^2*b*x^4 + 9*a*b^2*x^2 + 2*b^3 - 6*(a^3*x^6 + 2*a^2*b*x^4 + a*b^2*x^2)*
log(a*x^2 + b) + 12*(a^3*x^6 + 2*a^2*b*x^4 + a*b^2*x^2)*log(x))/(a^2*b^4*x^6 + 2
*a*b^5*x^4 + b^6*x^2)

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Sympy [A]  time = 3.01482, size = 78, normalized size = 1.16 \[ - \frac{3 a \log{\left (x \right )}}{b^{4}} + \frac{3 a \log{\left (x^{2} + \frac{b}{a} \right )}}{2 b^{4}} - \frac{6 a^{2} x^{4} + 9 a b x^{2} + 2 b^{2}}{4 a^{2} b^{3} x^{6} + 8 a b^{4} x^{4} + 4 b^{5} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**2)**3/x**9,x)

[Out]

-3*a*log(x)/b**4 + 3*a*log(x**2 + b/a)/(2*b**4) - (6*a**2*x**4 + 9*a*b*x**2 + 2*
b**2)/(4*a**2*b**3*x**6 + 8*a*b**4*x**4 + 4*b**5*x**2)

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GIAC/XCAS [A]  time = 0.235527, size = 111, normalized size = 1.66 \[ -\frac{3 \, a{\rm ln}\left (x^{2}\right )}{2 \, b^{4}} + \frac{3 \, a{\rm ln}\left ({\left | a x^{2} + b \right |}\right )}{2 \, b^{4}} - \frac{9 \, a^{3} x^{4} + 22 \, a^{2} b x^{2} + 14 \, a b^{2}}{4 \,{\left (a x^{2} + b\right )}^{2} b^{4}} + \frac{3 \, a x^{2} - b}{2 \, b^{4} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^3*x^9),x, algorithm="giac")

[Out]

-3/2*a*ln(x^2)/b^4 + 3/2*a*ln(abs(a*x^2 + b))/b^4 - 1/4*(9*a^3*x^4 + 22*a^2*b*x^
2 + 14*a*b^2)/((a*x^2 + b)^2*b^4) + 1/2*(3*a*x^2 - b)/(b^4*x^2)