3.1869 \(\int \frac{1}{\left (a+\frac{b}{x^2}\right )^2 x^7} \, dx\)

Optimal. Leaf size=49 \[ \frac{a \log \left (a x^2+b\right )}{b^3}-\frac{2 a \log (x)}{b^3}-\frac{a}{2 b^2 \left (a x^2+b\right )}-\frac{1}{2 b^2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.0978796, antiderivative size = 49, 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 \[ \frac{a \log \left (a x^2+b\right )}{b^3}-\frac{2 a \log (x)}{b^3}-\frac{a}{2 b^2 \left (a x^2+b\right )}-\frac{1}{2 b^2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^2)^2*x^7),x]

[Out]

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

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Rubi in Sympy [A]  time = 12.1503, size = 46, normalized size = 0.94 \[ - \frac{a}{2 b^{2} \left (a x^{2} + b\right )} - \frac{a \log{\left (x^{2} \right )}}{b^{3}} + \frac{a \log{\left (a x^{2} + b \right )}}{b^{3}} - \frac{1}{2 b^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0633077, size = 41, normalized size = 0.84 \[ -\frac{b \left (\frac{a}{a x^2+b}+\frac{1}{x^2}\right )-2 a \log \left (a x^2+b\right )+4 a \log (x)}{2 b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.019, size = 46, normalized size = 0.9 \[ -{\frac{1}{2\,{b}^{2}{x}^{2}}}-{\frac{a}{2\,{b}^{2} \left ( a{x}^{2}+b \right ) }}-2\,{\frac{a\ln \left ( x \right ) }{{b}^{3}}}+{\frac{a\ln \left ( a{x}^{2}+b \right ) }{{b}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)^2/x^7,x)

[Out]

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

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Maxima [A]  time = 1.44481, size = 70, normalized size = 1.43 \[ -\frac{2 \, a x^{2} + b}{2 \,{\left (a b^{2} x^{4} + b^{3} x^{2}\right )}} + \frac{a \log \left (a x^{2} + b\right )}{b^{3}} - \frac{a \log \left (x^{2}\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.224445, size = 99, normalized size = 2.02 \[ -\frac{2 \, a b x^{2} + b^{2} - 2 \,{\left (a^{2} x^{4} + a b x^{2}\right )} \log \left (a x^{2} + b\right ) + 4 \,{\left (a^{2} x^{4} + a b x^{2}\right )} \log \left (x\right )}{2 \,{\left (a b^{3} x^{4} + b^{4} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.07918, size = 49, normalized size = 1. \[ - \frac{2 a \log{\left (x \right )}}{b^{3}} + \frac{a \log{\left (x^{2} + \frac{b}{a} \right )}}{b^{3}} - \frac{2 a x^{2} + b}{2 a b^{2} x^{4} + 2 b^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.226984, size = 69, normalized size = 1.41 \[ -\frac{a{\rm ln}\left (x^{2}\right )}{b^{3}} + \frac{a{\rm ln}\left ({\left | a x^{2} + b \right |}\right )}{b^{3}} - \frac{2 \, a x^{2} + b}{2 \,{\left (a x^{4} + b x^{2}\right )} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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