3.483 \(\int \frac{1}{x^5 \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.120071, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{3 b^2 \log \left (a+b x^2\right )}{2 a^4}+\frac{3 b^2 \log (x)}{a^4}+\frac{b^2}{2 a^3 \left (a+b x^2\right )}+\frac{b}{a^3 x^2}-\frac{1}{4 a^2 x^4} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*(a^2 + 2*a*b*x^2 + b^2*x^4)),x]

[Out]

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

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Rubi in Sympy [A]  time = 22.2745, size = 66, normalized size = 1. \[ - \frac{1}{4 a^{2} x^{4}} + \frac{b^{2}}{2 a^{3} \left (a + b x^{2}\right )} + \frac{b}{a^{3} x^{2}} + \frac{3 b^{2} \log{\left (x^{2} \right )}}{2 a^{4}} - \frac{3 b^{2} \log{\left (a + b x^{2} \right )}}{2 a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**5/(b**2*x**4+2*a*b*x**2+a**2),x)

[Out]

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

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Mathematica [A]  time = 0.117276, size = 57, normalized size = 0.86 \[ \frac{-6 b^2 \log \left (a+b x^2\right )+a \left (\frac{2 b^2}{a+b x^2}-\frac{a}{x^4}+\frac{4 b}{x^2}\right )+12 b^2 \log (x)}{4 a^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^5*(a^2 + 2*a*b*x^2 + b^2*x^4)),x]

[Out]

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

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Maple [A]  time = 0.02, size = 61, normalized size = 0.9 \[ -{\frac{1}{4\,{a}^{2}{x}^{4}}}+{\frac{b}{{a}^{3}{x}^{2}}}+{\frac{{b}^{2}}{2\,{a}^{3} \left ( b{x}^{2}+a \right ) }}+3\,{\frac{{b}^{2}\ln \left ( x \right ) }{{a}^{4}}}-{\frac{3\,{b}^{2}\ln \left ( b{x}^{2}+a \right ) }{2\,{a}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(b^2*x^4+2*a*b*x^2+a^2),x)

[Out]

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

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Maxima [A]  time = 0.689011, size = 95, normalized size = 1.44 \[ \frac{6 \, b^{2} x^{4} + 3 \, a b x^{2} - a^{2}}{4 \,{\left (a^{3} b x^{6} + a^{4} x^{4}\right )}} - \frac{3 \, b^{2} \log \left (b x^{2} + a\right )}{2 \, a^{4}} + \frac{3 \, b^{2} \log \left (x^{2}\right )}{2 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^4 + 2*a*b*x^2 + a^2)*x^5),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.259477, size = 122, normalized size = 1.85 \[ \frac{6 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} - a^{3} - 6 \,{\left (b^{3} x^{6} + a b^{2} x^{4}\right )} \log \left (b x^{2} + a\right ) + 12 \,{\left (b^{3} x^{6} + a b^{2} x^{4}\right )} \log \left (x\right )}{4 \,{\left (a^{4} b x^{6} + a^{5} x^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^4 + 2*a*b*x^2 + a^2)*x^5),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.42986, size = 68, normalized size = 1.03 \[ \frac{- a^{2} + 3 a b x^{2} + 6 b^{2} x^{4}}{4 a^{4} x^{4} + 4 a^{3} b x^{6}} + \frac{3 b^{2} \log{\left (x \right )}}{a^{4}} - \frac{3 b^{2} \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**5/(b**2*x**4+2*a*b*x**2+a**2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.269886, size = 116, normalized size = 1.76 \[ \frac{3 \, b^{2}{\rm ln}\left (x^{2}\right )}{2 \, a^{4}} - \frac{3 \, b^{2}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{4}} + \frac{3 \, b^{3} x^{2} + 4 \, a b^{2}}{2 \,{\left (b x^{2} + a\right )} a^{4}} - \frac{9 \, b^{2} x^{4} - 4 \, a b x^{2} + a^{2}}{4 \, a^{4} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^4 + 2*a*b*x^2 + a^2)*x^5),x, algorithm="giac")

[Out]

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