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

Optimal. Leaf size=116 \[ \frac{3 b \log \left (a+b x^2\right )}{a^7}-\frac{6 b \log (x)}{a^7}-\frac{5 b}{2 a^6 \left (a+b x^2\right )}-\frac{1}{2 a^6 x^2}-\frac{b}{a^5 \left (a+b x^2\right )^2}-\frac{b}{2 a^4 \left (a+b x^2\right )^3}-\frac{b}{4 a^3 \left (a+b x^2\right )^4}-\frac{b}{10 a^2 \left (a+b x^2\right )^5} \]

[Out]

-1/(2*a^6*x^2) - b/(10*a^2*(a + b*x^2)^5) - b/(4*a^3*(a + b*x^2)^4) - b/(2*a^4*(
a + b*x^2)^3) - b/(a^5*(a + b*x^2)^2) - (5*b)/(2*a^6*(a + b*x^2)) - (6*b*Log[x])
/a^7 + (3*b*Log[a + b*x^2])/a^7

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Rubi [A]  time = 0.28339, antiderivative size = 116, 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 \[ \frac{3 b \log \left (a+b x^2\right )}{a^7}-\frac{6 b \log (x)}{a^7}-\frac{5 b}{2 a^6 \left (a+b x^2\right )}-\frac{1}{2 a^6 x^2}-\frac{b}{a^5 \left (a+b x^2\right )^2}-\frac{b}{2 a^4 \left (a+b x^2\right )^3}-\frac{b}{4 a^3 \left (a+b x^2\right )^4}-\frac{b}{10 a^2 \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

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

[Out]

-1/(2*a^6*x^2) - b/(10*a^2*(a + b*x^2)^5) - b/(4*a^3*(a + b*x^2)^4) - b/(2*a^4*(
a + b*x^2)^3) - b/(a^5*(a + b*x^2)^2) - (5*b)/(2*a^6*(a + b*x^2)) - (6*b*Log[x])
/a^7 + (3*b*Log[a + b*x^2])/a^7

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Rubi in Sympy [A]  time = 43.2684, size = 110, normalized size = 0.95 \[ - \frac{b}{10 a^{2} \left (a + b x^{2}\right )^{5}} - \frac{b}{4 a^{3} \left (a + b x^{2}\right )^{4}} - \frac{b}{2 a^{4} \left (a + b x^{2}\right )^{3}} - \frac{b}{a^{5} \left (a + b x^{2}\right )^{2}} - \frac{5 b}{2 a^{6} \left (a + b x^{2}\right )} - \frac{1}{2 a^{6} x^{2}} - \frac{3 b \log{\left (x^{2} \right )}}{a^{7}} + \frac{3 b \log{\left (a + b x^{2} \right )}}{a^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b/(10*a**2*(a + b*x**2)**5) - b/(4*a**3*(a + b*x**2)**4) - b/(2*a**4*(a + b*x**
2)**3) - b/(a**5*(a + b*x**2)**2) - 5*b/(2*a**6*(a + b*x**2)) - 1/(2*a**6*x**2)
- 3*b*log(x**2)/a**7 + 3*b*log(a + b*x**2)/a**7

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Mathematica [A]  time = 0.154105, size = 92, normalized size = 0.79 \[ -\frac{\frac{a \left (10 a^5+137 a^4 b x^2+385 a^3 b^2 x^4+470 a^2 b^3 x^6+270 a b^4 x^8+60 b^5 x^{10}\right )}{x^2 \left (a+b x^2\right )^5}-60 b \log \left (a+b x^2\right )+120 b \log (x)}{20 a^7} \]

Antiderivative was successfully verified.

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

[Out]

-((a*(10*a^5 + 137*a^4*b*x^2 + 385*a^3*b^2*x^4 + 470*a^2*b^3*x^6 + 270*a*b^4*x^8
+ 60*b^5*x^10))/(x^2*(a + b*x^2)^5) + 120*b*Log[x] - 60*b*Log[a + b*x^2])/(20*a
^7)

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Maple [A]  time = 0.024, size = 107, normalized size = 0.9 \[ -{\frac{1}{2\,{a}^{6}{x}^{2}}}-{\frac{b}{10\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{5}}}-{\frac{b}{4\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{4}}}-{\frac{b}{2\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{3}}}-{\frac{b}{{a}^{5} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{5\,b}{2\,{a}^{6} \left ( b{x}^{2}+a \right ) }}-6\,{\frac{b\ln \left ( x \right ) }{{a}^{7}}}+3\,{\frac{b\ln \left ( b{x}^{2}+a \right ) }{{a}^{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/a^6/x^2-1/10*b/a^2/(b*x^2+a)^5-1/4*b/a^3/(b*x^2+a)^4-1/2*b/a^4/(b*x^2+a)^3-
b/a^5/(b*x^2+a)^2-5/2*b/a^6/(b*x^2+a)-6*b*ln(x)/a^7+3*b*ln(b*x^2+a)/a^7

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Maxima [A]  time = 0.699523, size = 193, normalized size = 1.66 \[ -\frac{60 \, b^{5} x^{10} + 270 \, a b^{4} x^{8} + 470 \, a^{2} b^{3} x^{6} + 385 \, a^{3} b^{2} x^{4} + 137 \, a^{4} b x^{2} + 10 \, a^{5}}{20 \,{\left (a^{6} b^{5} x^{12} + 5 \, a^{7} b^{4} x^{10} + 10 \, a^{8} b^{3} x^{8} + 10 \, a^{9} b^{2} x^{6} + 5 \, a^{10} b x^{4} + a^{11} x^{2}\right )}} + \frac{3 \, b \log \left (b x^{2} + a\right )}{a^{7}} - \frac{3 \, b \log \left (x^{2}\right )}{a^{7}} \]

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^3),x, algorithm="maxima")

[Out]

-1/20*(60*b^5*x^10 + 270*a*b^4*x^8 + 470*a^2*b^3*x^6 + 385*a^3*b^2*x^4 + 137*a^4
*b*x^2 + 10*a^5)/(a^6*b^5*x^12 + 5*a^7*b^4*x^10 + 10*a^8*b^3*x^8 + 10*a^9*b^2*x^
6 + 5*a^10*b*x^4 + a^11*x^2) + 3*b*log(b*x^2 + a)/a^7 - 3*b*log(x^2)/a^7

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Fricas [A]  time = 0.264739, size = 339, normalized size = 2.92 \[ -\frac{60 \, a b^{5} x^{10} + 270 \, a^{2} b^{4} x^{8} + 470 \, a^{3} b^{3} x^{6} + 385 \, a^{4} b^{2} x^{4} + 137 \, a^{5} b x^{2} + 10 \, a^{6} - 60 \,{\left (b^{6} x^{12} + 5 \, a b^{5} x^{10} + 10 \, a^{2} b^{4} x^{8} + 10 \, a^{3} b^{3} x^{6} + 5 \, a^{4} b^{2} x^{4} + a^{5} b x^{2}\right )} \log \left (b x^{2} + a\right ) + 120 \,{\left (b^{6} x^{12} + 5 \, a b^{5} x^{10} + 10 \, a^{2} b^{4} x^{8} + 10 \, a^{3} b^{3} x^{6} + 5 \, a^{4} b^{2} x^{4} + a^{5} b x^{2}\right )} \log \left (x\right )}{20 \,{\left (a^{7} b^{5} x^{12} + 5 \, a^{8} b^{4} x^{10} + 10 \, a^{9} b^{3} x^{8} + 10 \, a^{10} b^{2} x^{6} + 5 \, a^{11} b x^{4} + a^{12} x^{2}\right )}} \]

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^3),x, algorithm="fricas")

[Out]

-1/20*(60*a*b^5*x^10 + 270*a^2*b^4*x^8 + 470*a^3*b^3*x^6 + 385*a^4*b^2*x^4 + 137
*a^5*b*x^2 + 10*a^6 - 60*(b^6*x^12 + 5*a*b^5*x^10 + 10*a^2*b^4*x^8 + 10*a^3*b^3*
x^6 + 5*a^4*b^2*x^4 + a^5*b*x^2)*log(b*x^2 + a) + 120*(b^6*x^12 + 5*a*b^5*x^10 +
10*a^2*b^4*x^8 + 10*a^3*b^3*x^6 + 5*a^4*b^2*x^4 + a^5*b*x^2)*log(x))/(a^7*b^5*x
^12 + 5*a^8*b^4*x^10 + 10*a^9*b^3*x^8 + 10*a^10*b^2*x^6 + 5*a^11*b*x^4 + a^12*x^
2)

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Sympy [A]  time = 26.8549, size = 148, normalized size = 1.28 \[ - \frac{10 a^{5} + 137 a^{4} b x^{2} + 385 a^{3} b^{2} x^{4} + 470 a^{2} b^{3} x^{6} + 270 a b^{4} x^{8} + 60 b^{5} x^{10}}{20 a^{11} x^{2} + 100 a^{10} b x^{4} + 200 a^{9} b^{2} x^{6} + 200 a^{8} b^{3} x^{8} + 100 a^{7} b^{4} x^{10} + 20 a^{6} b^{5} x^{12}} - \frac{6 b \log{\left (x \right )}}{a^{7}} + \frac{3 b \log{\left (\frac{a}{b} + x^{2} \right )}}{a^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(10*a**5 + 137*a**4*b*x**2 + 385*a**3*b**2*x**4 + 470*a**2*b**3*x**6 + 270*a*b*
*4*x**8 + 60*b**5*x**10)/(20*a**11*x**2 + 100*a**10*b*x**4 + 200*a**9*b**2*x**6
+ 200*a**8*b**3*x**8 + 100*a**7*b**4*x**10 + 20*a**6*b**5*x**12) - 6*b*log(x)/a*
*7 + 3*b*log(a/b + x**2)/a**7

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GIAC/XCAS [A]  time = 0.272588, size = 155, normalized size = 1.34 \[ -\frac{3 \, b{\rm ln}\left (x^{2}\right )}{a^{7}} + \frac{3 \, b{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{a^{7}} + \frac{6 \, b x^{2} - a}{2 \, a^{7} x^{2}} - \frac{137 \, b^{6} x^{10} + 735 \, a b^{5} x^{8} + 1590 \, a^{2} b^{4} x^{6} + 1740 \, a^{3} b^{3} x^{4} + 970 \, a^{4} b^{2} x^{2} + 224 \, a^{5} b}{20 \,{\left (b x^{2} + a\right )}^{5} a^{7}} \]

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^3),x, algorithm="giac")

[Out]

-3*b*ln(x^2)/a^7 + 3*b*ln(abs(b*x^2 + a))/a^7 + 1/2*(6*b*x^2 - a)/(a^7*x^2) - 1/
20*(137*b^6*x^10 + 735*a*b^5*x^8 + 1590*a^2*b^4*x^6 + 1740*a^3*b^3*x^4 + 970*a^4
*b^2*x^2 + 224*a^5*b)/((b*x^2 + a)^5*a^7)