3.1888 \(\int \frac{1}{\left (a+\frac{b}{x^2}\right )^3 x^{11}} \, dx\)

Optimal. Leaf size=86 \[ -\frac{3 a^2 \log \left (a x^2+b\right )}{b^5}+\frac{6 a^2 \log (x)}{b^5}+\frac{3 a^2}{2 b^4 \left (a x^2+b\right )}+\frac{a^2}{4 b^3 \left (a x^2+b\right )^2}+\frac{3 a}{2 b^4 x^2}-\frac{1}{4 b^3 x^4} \]

[Out]

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

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Rubi [A]  time = 0.157385, antiderivative size = 86, 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 \[ -\frac{3 a^2 \log \left (a x^2+b\right )}{b^5}+\frac{6 a^2 \log (x)}{b^5}+\frac{3 a^2}{2 b^4 \left (a x^2+b\right )}+\frac{a^2}{4 b^3 \left (a x^2+b\right )^2}+\frac{3 a}{2 b^4 x^2}-\frac{1}{4 b^3 x^4} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^2)^3*x^11),x]

[Out]

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

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Rubi in Sympy [A]  time = 19.7589, size = 85, normalized size = 0.99 \[ \frac{a^{2}}{4 b^{3} \left (a x^{2} + b\right )^{2}} + \frac{3 a^{2}}{2 b^{4} \left (a x^{2} + b\right )} + \frac{3 a^{2} \log{\left (x^{2} \right )}}{b^{5}} - \frac{3 a^{2} \log{\left (a x^{2} + b \right )}}{b^{5}} + \frac{3 a}{2 b^{4} x^{2}} - \frac{1}{4 b^{3} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**2)**3/x**11,x)

[Out]

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

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Mathematica [A]  time = 0.0870056, size = 74, normalized size = 0.86 \[ \frac{-12 a^2 \log \left (a x^2+b\right )+24 a^2 \log (x)+\frac{b \left (12 a^3 x^6+18 a^2 b x^4+4 a b^2 x^2-b^3\right )}{x^4 \left (a x^2+b\right )^2}}{4 b^5} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x^2)^3*x^11),x]

[Out]

((b*(-b^3 + 4*a*b^2*x^2 + 18*a^2*b*x^4 + 12*a^3*x^6))/(x^4*(b + a*x^2)^2) + 24*a
^2*Log[x] - 12*a^2*Log[b + a*x^2])/(4*b^5)

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Maple [A]  time = 0.018, size = 79, normalized size = 0.9 \[ -{\frac{1}{4\,{b}^{3}{x}^{4}}}+{\frac{3\,a}{2\,{b}^{4}{x}^{2}}}+{\frac{{a}^{2}}{4\,{b}^{3} \left ( a{x}^{2}+b \right ) ^{2}}}+{\frac{3\,{a}^{2}}{2\,{b}^{4} \left ( a{x}^{2}+b \right ) }}+6\,{\frac{{a}^{2}\ln \left ( x \right ) }{{b}^{5}}}-3\,{\frac{{a}^{2}\ln \left ( a{x}^{2}+b \right ) }{{b}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)^3/x^11,x)

[Out]

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

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Maxima [A]  time = 1.43148, size = 124, normalized size = 1.44 \[ \frac{12 \, a^{3} x^{6} + 18 \, a^{2} b x^{4} + 4 \, a b^{2} x^{2} - b^{3}}{4 \,{\left (a^{2} b^{4} x^{8} + 2 \, a b^{5} x^{6} + b^{6} x^{4}\right )}} - \frac{3 \, a^{2} \log \left (a x^{2} + b\right )}{b^{5}} + \frac{3 \, a^{2} \log \left (x^{2}\right )}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^3*x^11),x, algorithm="maxima")

[Out]

1/4*(12*a^3*x^6 + 18*a^2*b*x^4 + 4*a*b^2*x^2 - b^3)/(a^2*b^4*x^8 + 2*a*b^5*x^6 +
b^6*x^4) - 3*a^2*log(a*x^2 + b)/b^5 + 3*a^2*log(x^2)/b^5

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Fricas [A]  time = 0.228568, size = 181, normalized size = 2.1 \[ \frac{12 \, a^{3} b x^{6} + 18 \, a^{2} b^{2} x^{4} + 4 \, a b^{3} x^{2} - b^{4} - 12 \,{\left (a^{4} x^{8} + 2 \, a^{3} b x^{6} + a^{2} b^{2} x^{4}\right )} \log \left (a x^{2} + b\right ) + 24 \,{\left (a^{4} x^{8} + 2 \, a^{3} b x^{6} + a^{2} b^{2} x^{4}\right )} \log \left (x\right )}{4 \,{\left (a^{2} b^{5} x^{8} + 2 \, a b^{6} x^{6} + b^{7} x^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^3*x^11),x, algorithm="fricas")

[Out]

1/4*(12*a^3*b*x^6 + 18*a^2*b^2*x^4 + 4*a*b^3*x^2 - b^4 - 12*(a^4*x^8 + 2*a^3*b*x
^6 + a^2*b^2*x^4)*log(a*x^2 + b) + 24*(a^4*x^8 + 2*a^3*b*x^6 + a^2*b^2*x^4)*log(
x))/(a^2*b^5*x^8 + 2*a*b^6*x^6 + b^7*x^4)

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Sympy [A]  time = 4.45709, size = 90, normalized size = 1.05 \[ \frac{6 a^{2} \log{\left (x \right )}}{b^{5}} - \frac{3 a^{2} \log{\left (x^{2} + \frac{b}{a} \right )}}{b^{5}} + \frac{12 a^{3} x^{6} + 18 a^{2} b x^{4} + 4 a b^{2} x^{2} - b^{3}}{4 a^{2} b^{4} x^{8} + 8 a b^{5} x^{6} + 4 b^{6} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**2)**3/x**11,x)

[Out]

6*a**2*log(x)/b**5 - 3*a**2*log(x**2 + b/a)/b**5 + (12*a**3*x**6 + 18*a**2*b*x**
4 + 4*a*b**2*x**2 - b**3)/(4*a**2*b**4*x**8 + 8*a*b**5*x**6 + 4*b**6*x**4)

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GIAC/XCAS [A]  time = 0.224282, size = 108, normalized size = 1.26 \[ \frac{3 \, a^{2}{\rm ln}\left (x^{2}\right )}{b^{5}} - \frac{3 \, a^{2}{\rm ln}\left ({\left | a x^{2} + b \right |}\right )}{b^{5}} + \frac{12 \, a^{3} x^{6} + 18 \, a^{2} b x^{4} + 4 \, a b^{2} x^{2} - b^{3}}{4 \,{\left (a x^{4} + b x^{2}\right )}^{2} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^3*x^11),x, algorithm="giac")

[Out]

3*a^2*ln(x^2)/b^5 - 3*a^2*ln(abs(a*x^2 + b))/b^5 + 1/4*(12*a^3*x^6 + 18*a^2*b*x^
4 + 4*a*b^2*x^2 - b^3)/((a*x^4 + b*x^2)^2*b^4)