3.1884 \(\int \frac{1}{\left (a+\frac{b}{x^2}\right )^3 x^7} \, dx\)

Optimal. Leaf size=54 \[ -\frac{\log \left (a x^2+b\right )}{2 b^3}+\frac{1}{2 b^2 \left (a x^2+b\right )}+\frac{1}{4 b \left (a x^2+b\right )^2}+\frac{\log (x)}{b^3} \]

[Out]

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

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Rubi [A]  time = 0.100204, antiderivative size = 54, 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 \[ -\frac{\log \left (a x^2+b\right )}{2 b^3}+\frac{1}{2 b^2 \left (a x^2+b\right )}+\frac{1}{4 b \left (a x^2+b\right )^2}+\frac{\log (x)}{b^3} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^2)^3*x^7),x]

[Out]

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

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Rubi in Sympy [A]  time = 12.7582, size = 49, normalized size = 0.91 \[ \frac{1}{4 b \left (a x^{2} + b\right )^{2}} + \frac{1}{2 b^{2} \left (a x^{2} + b\right )} + \frac{\log{\left (x^{2} \right )}}{2 b^{3}} - \frac{\log{\left (a x^{2} + b \right )}}{2 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**2)**3/x**7,x)

[Out]

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

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Mathematica [A]  time = 0.0530109, size = 43, normalized size = 0.8 \[ \frac{\frac{b \left (2 a x^2+3 b\right )}{\left (a x^2+b\right )^2}-2 \log \left (a x^2+b\right )+4 \log (x)}{4 b^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x^2)^3*x^7),x]

[Out]

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

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Maple [A]  time = 0.016, size = 49, normalized size = 0.9 \[{\frac{1}{4\,b \left ( a{x}^{2}+b \right ) ^{2}}}+{\frac{1}{2\,{b}^{2} \left ( a{x}^{2}+b \right ) }}+{\frac{\ln \left ( x \right ) }{{b}^{3}}}-{\frac{\ln \left ( a{x}^{2}+b \right ) }{2\,{b}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)^3/x^7,x)

[Out]

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

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Maxima [A]  time = 1.42213, size = 81, normalized size = 1.5 \[ \frac{2 \, a x^{2} + 3 \, b}{4 \,{\left (a^{2} b^{2} x^{4} + 2 \, a b^{3} x^{2} + b^{4}\right )}} - \frac{\log \left (a x^{2} + b\right )}{2 \, b^{3}} + \frac{\log \left (x^{2}\right )}{2 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^3*x^7),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.226441, size = 122, normalized size = 2.26 \[ \frac{2 \, a b x^{2} + 3 \, b^{2} - 2 \,{\left (a^{2} x^{4} + 2 \, a b x^{2} + b^{2}\right )} \log \left (a x^{2} + b\right ) + 4 \,{\left (a^{2} x^{4} + 2 \, a b x^{2} + b^{2}\right )} \log \left (x\right )}{4 \,{\left (a^{2} b^{3} x^{4} + 2 \, a b^{4} x^{2} + b^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^3*x^7),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.25035, size = 56, normalized size = 1.04 \[ \frac{2 a x^{2} + 3 b}{4 a^{2} b^{2} x^{4} + 8 a b^{3} x^{2} + 4 b^{4}} + \frac{\log{\left (x \right )}}{b^{3}} - \frac{\log{\left (x^{2} + \frac{b}{a} \right )}}{2 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**2)**3/x**7,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.226274, size = 80, normalized size = 1.48 \[ \frac{{\rm ln}\left (x^{2}\right )}{2 \, b^{3}} - \frac{{\rm ln}\left ({\left | a x^{2} + b \right |}\right )}{2 \, b^{3}} + \frac{3 \, a^{2} x^{4} + 8 \, a b x^{2} + 6 \, b^{2}}{4 \,{\left (a x^{2} + b\right )}^{2} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^3*x^7),x, algorithm="giac")

[Out]

1/2*ln(x^2)/b^3 - 1/2*ln(abs(a*x^2 + b))/b^3 + 1/4*(3*a^2*x^4 + 8*a*b*x^2 + 6*b^
2)/((a*x^2 + b)^2*b^3)