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

Optimal. Leaf size=70 \[ -\frac{\log \left (a+b x^2\right )}{2 a^4}+\frac{\log (x)}{a^4}+\frac{1}{2 a^3 \left (a+b x^2\right )}+\frac{1}{4 a^2 \left (a+b x^2\right )^2}+\frac{1}{6 a \left (a+b x^2\right )^3} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.155505, antiderivative size = 70, 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 \[ -\frac{\log \left (a+b x^2\right )}{2 a^4}+\frac{\log (x)}{a^4}+\frac{1}{2 a^3 \left (a+b x^2\right )}+\frac{1}{4 a^2 \left (a+b x^2\right )^2}+\frac{1}{6 a \left (a+b x^2\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 28.9016, size = 65, normalized size = 0.93 \[ \frac{1}{6 a \left (a + b x^{2}\right )^{3}} + \frac{1}{4 a^{2} \left (a + b x^{2}\right )^{2}} + \frac{1}{2 a^{3} \left (a + b x^{2}\right )} + \frac{\log{\left (x^{2} \right )}}{2 a^{4}} - \frac{\log{\left (a + b x^{2} \right )}}{2 a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0747477, size = 54, normalized size = 0.77 \[ \frac{\frac{a \left (11 a^2+15 a b x^2+6 b^2 x^4\right )}{\left (a+b x^2\right )^3}-6 \log \left (a+b x^2\right )+12 \log (x)}{12 a^4} \]

Antiderivative was successfully verified.

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

[Out]

((a*(11*a^2 + 15*a*b*x^2 + 6*b^2*x^4))/(a + b*x^2)^3 + 12*Log[x] - 6*Log[a + b*x
^2])/(12*a^4)

_______________________________________________________________________________________

Maple [A]  time = 0.018, size = 63, normalized size = 0.9 \[{\frac{1}{6\,a \left ( b{x}^{2}+a \right ) ^{3}}}+{\frac{1}{4\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{1}{2\,{a}^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{\ln \left ( x \right ) }{{a}^{4}}}-{\frac{\ln \left ( b{x}^{2}+a \right ) }{2\,{a}^{4}}} \]

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)^2,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.693224, size = 111, normalized size = 1.59 \[ \frac{6 \, b^{2} x^{4} + 15 \, a b x^{2} + 11 \, a^{2}}{12 \,{\left (a^{3} b^{3} x^{6} + 3 \, a^{4} b^{2} x^{4} + 3 \, a^{5} b x^{2} + a^{6}\right )}} - \frac{\log \left (b x^{2} + a\right )}{2 \, a^{4}} + \frac{\log \left (x^{2}\right )}{2 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(6*b^2*x^4 + 15*a*b*x^2 + 11*a^2)/(a^3*b^3*x^6 + 3*a^4*b^2*x^4 + 3*a^5*b*x^
2 + a^6) - 1/2*log(b*x^2 + a)/a^4 + 1/2*log(x^2)/a^4

_______________________________________________________________________________________

Fricas [A]  time = 0.269195, size = 181, normalized size = 2.59 \[ \frac{6 \, a b^{2} x^{4} + 15 \, a^{2} b x^{2} + 11 \, a^{3} - 6 \,{\left (b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}\right )} \log \left (b x^{2} + a\right ) + 12 \,{\left (b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}\right )} \log \left (x\right )}{12 \,{\left (a^{4} b^{3} x^{6} + 3 \, a^{5} b^{2} x^{4} + 3 \, a^{6} b x^{2} + a^{7}\right )}} \]

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),x, algorithm="fricas")

[Out]

1/12*(6*a*b^2*x^4 + 15*a^2*b*x^2 + 11*a^3 - 6*(b^3*x^6 + 3*a*b^2*x^4 + 3*a^2*b*x
^2 + a^3)*log(b*x^2 + a) + 12*(b^3*x^6 + 3*a*b^2*x^4 + 3*a^2*b*x^2 + a^3)*log(x)
)/(a^4*b^3*x^6 + 3*a^5*b^2*x^4 + 3*a^6*b*x^2 + a^7)

_______________________________________________________________________________________

Sympy [A]  time = 2.95668, size = 80, normalized size = 1.14 \[ \frac{11 a^{2} + 15 a b x^{2} + 6 b^{2} x^{4}}{12 a^{6} + 36 a^{5} b x^{2} + 36 a^{4} b^{2} x^{4} + 12 a^{3} b^{3} x^{6}} + \frac{\log{\left (x \right )}}{a^{4}} - \frac{\log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(11*a**2 + 15*a*b*x**2 + 6*b**2*x**4)/(12*a**6 + 36*a**5*b*x**2 + 36*a**4*b**2*x
**4 + 12*a**3*b**3*x**6) + log(x)/a**4 - log(a/b + x**2)/(2*a**4)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.270831, size = 95, normalized size = 1.36 \[ \frac{{\rm ln}\left (x^{2}\right )}{2 \, a^{4}} - \frac{{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{4}} + \frac{11 \, b^{3} x^{6} + 39 \, a b^{2} x^{4} + 48 \, a^{2} b x^{2} + 22 \, a^{3}}{12 \,{\left (b x^{2} + a\right )}^{3} a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*ln(x^2)/a^4 - 1/2*ln(abs(b*x^2 + a))/a^4 + 1/12*(11*b^3*x^6 + 39*a*b^2*x^4 +
48*a^2*b*x^2 + 22*a^3)/((b*x^2 + a)^3*a^4)