3.520 \(\int \frac{1}{x \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx\)

Optimal. Leaf size=102 \[ -\frac{\log \left (a+b x^2\right )}{2 a^6}+\frac{\log (x)}{a^6}+\frac{1}{2 a^5 \left (a+b x^2\right )}+\frac{1}{4 a^4 \left (a+b x^2\right )^2}+\frac{1}{6 a^3 \left (a+b x^2\right )^3}+\frac{1}{8 a^2 \left (a+b x^2\right )^4}+\frac{1}{10 a \left (a+b x^2\right )^5} \]

[Out]

1/(10*a*(a + b*x^2)^5) + 1/(8*a^2*(a + b*x^2)^4) + 1/(6*a^3*(a + b*x^2)^3) + 1/(
4*a^4*(a + b*x^2)^2) + 1/(2*a^5*(a + b*x^2)) + Log[x]/a^6 - Log[a + b*x^2]/(2*a^
6)

_______________________________________________________________________________________

Rubi [A]  time = 0.239805, antiderivative size = 102, 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 \[ -\frac{\log \left (a+b x^2\right )}{2 a^6}+\frac{\log (x)}{a^6}+\frac{1}{2 a^5 \left (a+b x^2\right )}+\frac{1}{4 a^4 \left (a+b x^2\right )^2}+\frac{1}{6 a^3 \left (a+b x^2\right )^3}+\frac{1}{8 a^2 \left (a+b x^2\right )^4}+\frac{1}{10 a \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a^2 + 2*a*b*x^2 + b^2*x^4)^3),x]

[Out]

1/(10*a*(a + b*x^2)^5) + 1/(8*a^2*(a + b*x^2)^4) + 1/(6*a^3*(a + b*x^2)^3) + 1/(
4*a^4*(a + b*x^2)^2) + 1/(2*a^5*(a + b*x^2)) + Log[x]/a^6 - Log[a + b*x^2]/(2*a^
6)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 35.9625, size = 95, normalized size = 0.93 \[ \frac{1}{10 a \left (a + b x^{2}\right )^{5}} + \frac{1}{8 a^{2} \left (a + b x^{2}\right )^{4}} + \frac{1}{6 a^{3} \left (a + b x^{2}\right )^{3}} + \frac{1}{4 a^{4} \left (a + b x^{2}\right )^{2}} + \frac{1}{2 a^{5} \left (a + b x^{2}\right )} + \frac{\log{\left (x^{2} \right )}}{2 a^{6}} - \frac{\log{\left (a + b x^{2} \right )}}{2 a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

1/(10*a*(a + b*x**2)**5) + 1/(8*a**2*(a + b*x**2)**4) + 1/(6*a**3*(a + b*x**2)**
3) + 1/(4*a**4*(a + b*x**2)**2) + 1/(2*a**5*(a + b*x**2)) + log(x**2)/(2*a**6) -
log(a + b*x**2)/(2*a**6)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0946772, size = 76, normalized size = 0.75 \[ \frac{\frac{a \left (137 a^4+385 a^3 b x^2+470 a^2 b^2 x^4+270 a b^3 x^6+60 b^4 x^8\right )}{\left (a+b x^2\right )^5}-60 \log \left (a+b x^2\right )+120 \log (x)}{120 a^6} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(a^2 + 2*a*b*x^2 + b^2*x^4)^3),x]

[Out]

((a*(137*a^4 + 385*a^3*b*x^2 + 470*a^2*b^2*x^4 + 270*a*b^3*x^6 + 60*b^4*x^8))/(a
+ b*x^2)^5 + 120*Log[x] - 60*Log[a + b*x^2])/(120*a^6)

_______________________________________________________________________________________

Maple [A]  time = 0.021, size = 91, normalized size = 0.9 \[{\frac{1}{10\,a \left ( b{x}^{2}+a \right ) ^{5}}}+{\frac{1}{8\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{4}}}+{\frac{1}{6\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{3}}}+{\frac{1}{4\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{1}{2\,{a}^{5} \left ( b{x}^{2}+a \right ) }}+{\frac{\ln \left ( x \right ) }{{a}^{6}}}-{\frac{\ln \left ( b{x}^{2}+a \right ) }{2\,{a}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

1/10/a/(b*x^2+a)^5+1/8/a^2/(b*x^2+a)^4+1/6/a^3/(b*x^2+a)^3+1/4/a^4/(b*x^2+a)^2+1
/2/a^5/(b*x^2+a)+ln(x)/a^6-1/2*ln(b*x^2+a)/a^6

_______________________________________________________________________________________

Maxima [A]  time = 0.693334, size = 170, normalized size = 1.67 \[ \frac{60 \, b^{4} x^{8} + 270 \, a b^{3} x^{6} + 470 \, a^{2} b^{2} x^{4} + 385 \, a^{3} b x^{2} + 137 \, a^{4}}{120 \,{\left (a^{5} b^{5} x^{10} + 5 \, a^{6} b^{4} x^{8} + 10 \, a^{7} b^{3} x^{6} + 10 \, a^{8} b^{2} x^{4} + 5 \, a^{9} b x^{2} + a^{10}\right )}} - \frac{\log \left (b x^{2} + a\right )}{2 \, a^{6}} + \frac{\log \left (x^{2}\right )}{2 \, a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^4 + 2*a*b*x^2 + a^2)^3*x),x, algorithm="maxima")

[Out]

1/120*(60*b^4*x^8 + 270*a*b^3*x^6 + 470*a^2*b^2*x^4 + 385*a^3*b*x^2 + 137*a^4)/(
a^5*b^5*x^10 + 5*a^6*b^4*x^8 + 10*a^7*b^3*x^6 + 10*a^8*b^2*x^4 + 5*a^9*b*x^2 + a
^10) - 1/2*log(b*x^2 + a)/a^6 + 1/2*log(x^2)/a^6

_______________________________________________________________________________________

Fricas [A]  time = 0.26622, size = 300, normalized size = 2.94 \[ \frac{60 \, a b^{4} x^{8} + 270 \, a^{2} b^{3} x^{6} + 470 \, a^{3} b^{2} x^{4} + 385 \, a^{4} b x^{2} + 137 \, a^{5} - 60 \,{\left (b^{5} x^{10} + 5 \, a b^{4} x^{8} + 10 \, a^{2} b^{3} x^{6} + 10 \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x^{2} + a^{5}\right )} \log \left (b x^{2} + a\right ) + 120 \,{\left (b^{5} x^{10} + 5 \, a b^{4} x^{8} + 10 \, a^{2} b^{3} x^{6} + 10 \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x^{2} + a^{5}\right )} \log \left (x\right )}{120 \,{\left (a^{6} b^{5} x^{10} + 5 \, a^{7} b^{4} x^{8} + 10 \, a^{8} b^{3} x^{6} + 10 \, a^{9} b^{2} x^{4} + 5 \, a^{10} b x^{2} + a^{11}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^4 + 2*a*b*x^2 + a^2)^3*x),x, algorithm="fricas")

[Out]

1/120*(60*a*b^4*x^8 + 270*a^2*b^3*x^6 + 470*a^3*b^2*x^4 + 385*a^4*b*x^2 + 137*a^
5 - 60*(b^5*x^10 + 5*a*b^4*x^8 + 10*a^2*b^3*x^6 + 10*a^3*b^2*x^4 + 5*a^4*b*x^2 +
a^5)*log(b*x^2 + a) + 120*(b^5*x^10 + 5*a*b^4*x^8 + 10*a^2*b^3*x^6 + 10*a^3*b^2
*x^4 + 5*a^4*b*x^2 + a^5)*log(x))/(a^6*b^5*x^10 + 5*a^7*b^4*x^8 + 10*a^8*b^3*x^6
+ 10*a^9*b^2*x^4 + 5*a^10*b*x^2 + a^11)

_______________________________________________________________________________________

Sympy [A]  time = 11.2931, size = 128, normalized size = 1.25 \[ \frac{137 a^{4} + 385 a^{3} b x^{2} + 470 a^{2} b^{2} x^{4} + 270 a b^{3} x^{6} + 60 b^{4} x^{8}}{120 a^{10} + 600 a^{9} b x^{2} + 1200 a^{8} b^{2} x^{4} + 1200 a^{7} b^{3} x^{6} + 600 a^{6} b^{4} x^{8} + 120 a^{5} b^{5} x^{10}} + \frac{\log{\left (x \right )}}{a^{6}} - \frac{\log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

(137*a**4 + 385*a**3*b*x**2 + 470*a**2*b**2*x**4 + 270*a*b**3*x**6 + 60*b**4*x**
8)/(120*a**10 + 600*a**9*b*x**2 + 1200*a**8*b**2*x**4 + 1200*a**7*b**3*x**6 + 60
0*a**6*b**4*x**8 + 120*a**5*b**5*x**10) + log(x)/a**6 - log(a/b + x**2)/(2*a**6)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.273273, size = 124, normalized size = 1.22 \[ \frac{{\rm ln}\left (x^{2}\right )}{2 \, a^{6}} - \frac{{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{6}} + \frac{137 \, b^{5} x^{10} + 745 \, a b^{4} x^{8} + 1640 \, a^{2} b^{3} x^{6} + 1840 \, a^{3} b^{2} x^{4} + 1070 \, a^{4} b x^{2} + 274 \, a^{5}}{120 \,{\left (b x^{2} + a\right )}^{5} a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^4 + 2*a*b*x^2 + a^2)^3*x),x, algorithm="giac")

[Out]

1/2*ln(x^2)/a^6 - 1/2*ln(abs(b*x^2 + a))/a^6 + 1/120*(137*b^5*x^10 + 745*a*b^4*x
^8 + 1640*a^2*b^3*x^6 + 1840*a^3*b^2*x^4 + 1070*a^4*b*x^2 + 274*a^5)/((b*x^2 + a
)^5*a^6)