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

Optimal. Leaf size=101 \[ -\frac{5 b^2 \log \left (a+b x^2\right )}{a^6}+\frac{10 b^2 \log (x)}{a^6}+\frac{3 b^2}{a^5 \left (a+b x^2\right )}+\frac{2 b}{a^5 x^2}+\frac{3 b^2}{4 a^4 \left (a+b x^2\right )^2}-\frac{1}{4 a^4 x^4}+\frac{b^2}{6 a^3 \left (a+b x^2\right )^3} \]

[Out]

-1/(4*a^4*x^4) + (2*b)/(a^5*x^2) + b^2/(6*a^3*(a + b*x^2)^3) + (3*b^2)/(4*a^4*(a
+ b*x^2)^2) + (3*b^2)/(a^5*(a + b*x^2)) + (10*b^2*Log[x])/a^6 - (5*b^2*Log[a +
b*x^2])/a^6

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Rubi [A]  time = 0.215025, antiderivative size = 101, 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 \[ -\frac{5 b^2 \log \left (a+b x^2\right )}{a^6}+\frac{10 b^2 \log (x)}{a^6}+\frac{3 b^2}{a^5 \left (a+b x^2\right )}+\frac{2 b}{a^5 x^2}+\frac{3 b^2}{4 a^4 \left (a+b x^2\right )^2}-\frac{1}{4 a^4 x^4}+\frac{b^2}{6 a^3 \left (a+b x^2\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

-1/(4*a^4*x^4) + (2*b)/(a^5*x^2) + b^2/(6*a^3*(a + b*x^2)^3) + (3*b^2)/(4*a^4*(a
+ b*x^2)^2) + (3*b^2)/(a^5*(a + b*x^2)) + (10*b^2*Log[x])/a^6 - (5*b^2*Log[a +
b*x^2])/a^6

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Rubi in Sympy [A]  time = 37.915, size = 100, normalized size = 0.99 \[ \frac{b^{2}}{6 a^{3} \left (a + b x^{2}\right )^{3}} + \frac{3 b^{2}}{4 a^{4} \left (a + b x^{2}\right )^{2}} - \frac{1}{4 a^{4} x^{4}} + \frac{3 b^{2}}{a^{5} \left (a + b x^{2}\right )} + \frac{2 b}{a^{5} x^{2}} + \frac{5 b^{2} \log{\left (x^{2} \right )}}{a^{6}} - \frac{5 b^{2} \log{\left (a + b x^{2} \right )}}{a^{6}} \]

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)**2,x)

[Out]

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

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Mathematica [A]  time = 0.108827, size = 85, normalized size = 0.84 \[ \frac{\frac{a \left (-3 a^4+15 a^3 b x^2+110 a^2 b^2 x^4+150 a b^3 x^6+60 b^4 x^8\right )}{x^4 \left (a+b x^2\right )^3}-60 b^2 \log \left (a+b x^2\right )+120 b^2 \log (x)}{12 a^6} \]

Antiderivative was successfully verified.

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

[Out]

((a*(-3*a^4 + 15*a^3*b*x^2 + 110*a^2*b^2*x^4 + 150*a*b^3*x^6 + 60*b^4*x^8))/(x^4
*(a + b*x^2)^3) + 120*b^2*Log[x] - 60*b^2*Log[a + b*x^2])/(12*a^6)

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Maple [A]  time = 0.022, size = 96, normalized size = 1. \[ -{\frac{1}{4\,{a}^{4}{x}^{4}}}+2\,{\frac{b}{{a}^{5}{x}^{2}}}+{\frac{{b}^{2}}{6\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{3}}}+{\frac{3\,{b}^{2}}{4\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}+3\,{\frac{{b}^{2}}{{a}^{5} \left ( b{x}^{2}+a \right ) }}+10\,{\frac{{b}^{2}\ln \left ( x \right ) }{{a}^{6}}}-5\,{\frac{{b}^{2}\ln \left ( b{x}^{2}+a \right ) }{{a}^{6}}} \]

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)^2,x)

[Out]

-1/4/a^4/x^4+2*b/a^5/x^2+1/6*b^2/a^3/(b*x^2+a)^3+3/4*b^2/a^4/(b*x^2+a)^2+3*b^2/a
^5/(b*x^2+a)+10*b^2*ln(x)/a^6-5*b^2*ln(b*x^2+a)/a^6

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Maxima [A]  time = 0.696789, size = 154, normalized size = 1.52 \[ \frac{60 \, b^{4} x^{8} + 150 \, a b^{3} x^{6} + 110 \, a^{2} b^{2} x^{4} + 15 \, a^{3} b x^{2} - 3 \, a^{4}}{12 \,{\left (a^{5} b^{3} x^{10} + 3 \, a^{6} b^{2} x^{8} + 3 \, a^{7} b x^{6} + a^{8} x^{4}\right )}} - \frac{5 \, b^{2} \log \left (b x^{2} + a\right )}{a^{6}} + \frac{5 \, b^{2} \log \left (x^{2}\right )}{a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(60*b^4*x^8 + 150*a*b^3*x^6 + 110*a^2*b^2*x^4 + 15*a^3*b*x^2 - 3*a^4)/(a^5*
b^3*x^10 + 3*a^6*b^2*x^8 + 3*a^7*b*x^6 + a^8*x^4) - 5*b^2*log(b*x^2 + a)/a^6 + 5
*b^2*log(x^2)/a^6

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Fricas [A]  time = 0.271841, size = 240, normalized size = 2.38 \[ \frac{60 \, a b^{4} x^{8} + 150 \, a^{2} b^{3} x^{6} + 110 \, a^{3} b^{2} x^{4} + 15 \, a^{4} b x^{2} - 3 \, a^{5} - 60 \,{\left (b^{5} x^{10} + 3 \, a b^{4} x^{8} + 3 \, a^{2} b^{3} x^{6} + a^{3} b^{2} x^{4}\right )} \log \left (b x^{2} + a\right ) + 120 \,{\left (b^{5} x^{10} + 3 \, a b^{4} x^{8} + 3 \, a^{2} b^{3} x^{6} + a^{3} b^{2} x^{4}\right )} \log \left (x\right )}{12 \,{\left (a^{6} b^{3} x^{10} + 3 \, a^{7} b^{2} x^{8} + 3 \, a^{8} b x^{6} + a^{9} x^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(60*a*b^4*x^8 + 150*a^2*b^3*x^6 + 110*a^3*b^2*x^4 + 15*a^4*b*x^2 - 3*a^5 -
60*(b^5*x^10 + 3*a*b^4*x^8 + 3*a^2*b^3*x^6 + a^3*b^2*x^4)*log(b*x^2 + a) + 120*(
b^5*x^10 + 3*a*b^4*x^8 + 3*a^2*b^3*x^6 + a^3*b^2*x^4)*log(x))/(a^6*b^3*x^10 + 3*
a^7*b^2*x^8 + 3*a^8*b*x^6 + a^9*x^4)

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Sympy [A]  time = 9.00198, size = 116, normalized size = 1.15 \[ \frac{- 3 a^{4} + 15 a^{3} b x^{2} + 110 a^{2} b^{2} x^{4} + 150 a b^{3} x^{6} + 60 b^{4} x^{8}}{12 a^{8} x^{4} + 36 a^{7} b x^{6} + 36 a^{6} b^{2} x^{8} + 12 a^{5} b^{3} x^{10}} + \frac{10 b^{2} \log{\left (x \right )}}{a^{6}} - \frac{5 b^{2} \log{\left (\frac{a}{b} + x^{2} \right )}}{a^{6}} \]

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)**2,x)

[Out]

(-3*a**4 + 15*a**3*b*x**2 + 110*a**2*b**2*x**4 + 150*a*b**3*x**6 + 60*b**4*x**8)
/(12*a**8*x**4 + 36*a**7*b*x**6 + 36*a**6*b**2*x**8 + 12*a**5*b**3*x**10) + 10*b
**2*log(x)/a**6 - 5*b**2*log(a/b + x**2)/a**6

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GIAC/XCAS [A]  time = 0.27134, size = 146, normalized size = 1.45 \[ \frac{5 \, b^{2}{\rm ln}\left (x^{2}\right )}{a^{6}} - \frac{5 \, b^{2}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{a^{6}} + \frac{110 \, b^{5} x^{6} + 366 \, a b^{4} x^{4} + 411 \, a^{2} b^{3} x^{2} + 157 \, a^{3} b^{2}}{12 \,{\left (b x^{2} + a\right )}^{3} a^{6}} - \frac{30 \, b^{2} x^{4} - 8 \, a b x^{2} + a^{2}}{4 \, a^{6} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*b^2*ln(x^2)/a^6 - 5*b^2*ln(abs(b*x^2 + a))/a^6 + 1/12*(110*b^5*x^6 + 366*a*b^4
*x^4 + 411*a^2*b^3*x^2 + 157*a^3*b^2)/((b*x^2 + a)^3*a^6) - 1/4*(30*b^2*x^4 - 8*
a*b*x^2 + a^2)/(a^6*x^4)