3.522 \(\int \frac{1}{x^5 \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx\)

Optimal. Leaf size=140 \[ -\frac{21 b^2 \log \left (a+b x^2\right )}{2 a^8}+\frac{21 b^2 \log (x)}{a^8}+\frac{15 b^2}{2 a^7 \left (a+b x^2\right )}+\frac{3 b}{a^7 x^2}+\frac{5 b^2}{2 a^6 \left (a+b x^2\right )^2}-\frac{1}{4 a^6 x^4}+\frac{b^2}{a^5 \left (a+b x^2\right )^3}+\frac{3 b^2}{8 a^4 \left (a+b x^2\right )^4}+\frac{b^2}{10 a^3 \left (a+b x^2\right )^5} \]

[Out]

-1/(4*a^6*x^4) + (3*b)/(a^7*x^2) + b^2/(10*a^3*(a + b*x^2)^5) + (3*b^2)/(8*a^4*(
a + b*x^2)^4) + b^2/(a^5*(a + b*x^2)^3) + (5*b^2)/(2*a^6*(a + b*x^2)^2) + (15*b^
2)/(2*a^7*(a + b*x^2)) + (21*b^2*Log[x])/a^8 - (21*b^2*Log[a + b*x^2])/(2*a^8)

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Rubi [A]  time = 0.31955, antiderivative size = 140, 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 \[ -\frac{21 b^2 \log \left (a+b x^2\right )}{2 a^8}+\frac{21 b^2 \log (x)}{a^8}+\frac{15 b^2}{2 a^7 \left (a+b x^2\right )}+\frac{3 b}{a^7 x^2}+\frac{5 b^2}{2 a^6 \left (a+b x^2\right )^2}-\frac{1}{4 a^6 x^4}+\frac{b^2}{a^5 \left (a+b x^2\right )^3}+\frac{3 b^2}{8 a^4 \left (a+b x^2\right )^4}+\frac{b^2}{10 a^3 \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*(a^2 + 2*a*b*x^2 + b^2*x^4)^3),x]

[Out]

-1/(4*a^6*x^4) + (3*b)/(a^7*x^2) + b^2/(10*a^3*(a + b*x^2)^5) + (3*b^2)/(8*a^4*(
a + b*x^2)^4) + b^2/(a^5*(a + b*x^2)^3) + (5*b^2)/(2*a^6*(a + b*x^2)^2) + (15*b^
2)/(2*a^7*(a + b*x^2)) + (21*b^2*Log[x])/a^8 - (21*b^2*Log[a + b*x^2])/(2*a^8)

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Rubi in Sympy [A]  time = 56.7045, size = 139, normalized size = 0.99 \[ \frac{b^{2}}{10 a^{3} \left (a + b x^{2}\right )^{5}} + \frac{3 b^{2}}{8 a^{4} \left (a + b x^{2}\right )^{4}} + \frac{b^{2}}{a^{5} \left (a + b x^{2}\right )^{3}} + \frac{5 b^{2}}{2 a^{6} \left (a + b x^{2}\right )^{2}} - \frac{1}{4 a^{6} x^{4}} + \frac{15 b^{2}}{2 a^{7} \left (a + b x^{2}\right )} + \frac{3 b}{a^{7} x^{2}} + \frac{21 b^{2} \log{\left (x^{2} \right )}}{2 a^{8}} - \frac{21 b^{2} \log{\left (a + b x^{2} \right )}}{2 a^{8}} \]

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)**3,x)

[Out]

b**2/(10*a**3*(a + b*x**2)**5) + 3*b**2/(8*a**4*(a + b*x**2)**4) + b**2/(a**5*(a
+ b*x**2)**3) + 5*b**2/(2*a**6*(a + b*x**2)**2) - 1/(4*a**6*x**4) + 15*b**2/(2*
a**7*(a + b*x**2)) + 3*b/(a**7*x**2) + 21*b**2*log(x**2)/(2*a**8) - 21*b**2*log(
a + b*x**2)/(2*a**8)

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Mathematica [A]  time = 0.109802, size = 107, normalized size = 0.76 \[ \frac{\frac{a \left (-10 a^6+70 a^5 b x^2+959 a^4 b^2 x^4+2695 a^3 b^3 x^6+3290 a^2 b^4 x^8+1890 a b^5 x^{10}+420 b^6 x^{12}\right )}{x^4 \left (a+b x^2\right )^5}-420 b^2 \log \left (a+b x^2\right )+840 b^2 \log (x)}{40 a^8} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^5*(a^2 + 2*a*b*x^2 + b^2*x^4)^3),x]

[Out]

((a*(-10*a^6 + 70*a^5*b*x^2 + 959*a^4*b^2*x^4 + 2695*a^3*b^3*x^6 + 3290*a^2*b^4*
x^8 + 1890*a*b^5*x^10 + 420*b^6*x^12))/(x^4*(a + b*x^2)^5) + 840*b^2*Log[x] - 42
0*b^2*Log[a + b*x^2])/(40*a^8)

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Maple [A]  time = 0.025, size = 129, normalized size = 0.9 \[ -{\frac{1}{4\,{a}^{6}{x}^{4}}}+3\,{\frac{b}{{a}^{7}{x}^{2}}}+{\frac{{b}^{2}}{10\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{5}}}+{\frac{3\,{b}^{2}}{8\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{4}}}+{\frac{{b}^{2}}{{a}^{5} \left ( b{x}^{2}+a \right ) ^{3}}}+{\frac{5\,{b}^{2}}{2\,{a}^{6} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{15\,{b}^{2}}{2\,{a}^{7} \left ( b{x}^{2}+a \right ) }}+21\,{\frac{{b}^{2}\ln \left ( x \right ) }{{a}^{8}}}-{\frac{21\,{b}^{2}\ln \left ( b{x}^{2}+a \right ) }{2\,{a}^{8}}} \]

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)^3,x)

[Out]

-1/4/a^6/x^4+3*b/a^7/x^2+1/10*b^2/a^3/(b*x^2+a)^5+3/8*b^2/a^4/(b*x^2+a)^4+b^2/a^
5/(b*x^2+a)^3+5/2*b^2/a^6/(b*x^2+a)^2+15/2*b^2/a^7/(b*x^2+a)+21*b^2*ln(x)/a^8-21
/2*b^2*ln(b*x^2+a)/a^8

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Maxima [A]  time = 0.705082, size = 213, normalized size = 1.52 \[ \frac{420 \, b^{6} x^{12} + 1890 \, a b^{5} x^{10} + 3290 \, a^{2} b^{4} x^{8} + 2695 \, a^{3} b^{3} x^{6} + 959 \, a^{4} b^{2} x^{4} + 70 \, a^{5} b x^{2} - 10 \, a^{6}}{40 \,{\left (a^{7} b^{5} x^{14} + 5 \, a^{8} b^{4} x^{12} + 10 \, a^{9} b^{3} x^{10} + 10 \, a^{10} b^{2} x^{8} + 5 \, a^{11} b x^{6} + a^{12} x^{4}\right )}} - \frac{21 \, b^{2} \log \left (b x^{2} + a\right )}{2 \, a^{8}} + \frac{21 \, b^{2} \log \left (x^{2}\right )}{2 \, a^{8}} \]

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^5),x, algorithm="maxima")

[Out]

1/40*(420*b^6*x^12 + 1890*a*b^5*x^10 + 3290*a^2*b^4*x^8 + 2695*a^3*b^3*x^6 + 959
*a^4*b^2*x^4 + 70*a^5*b*x^2 - 10*a^6)/(a^7*b^5*x^14 + 5*a^8*b^4*x^12 + 10*a^9*b^
3*x^10 + 10*a^10*b^2*x^8 + 5*a^11*b*x^6 + a^12*x^4) - 21/2*b^2*log(b*x^2 + a)/a^
8 + 21/2*b^2*log(x^2)/a^8

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Fricas [A]  time = 0.263669, size = 359, normalized size = 2.56 \[ \frac{420 \, a b^{6} x^{12} + 1890 \, a^{2} b^{5} x^{10} + 3290 \, a^{3} b^{4} x^{8} + 2695 \, a^{4} b^{3} x^{6} + 959 \, a^{5} b^{2} x^{4} + 70 \, a^{6} b x^{2} - 10 \, a^{7} - 420 \,{\left (b^{7} x^{14} + 5 \, a b^{6} x^{12} + 10 \, a^{2} b^{5} x^{10} + 10 \, a^{3} b^{4} x^{8} + 5 \, a^{4} b^{3} x^{6} + a^{5} b^{2} x^{4}\right )} \log \left (b x^{2} + a\right ) + 840 \,{\left (b^{7} x^{14} + 5 \, a b^{6} x^{12} + 10 \, a^{2} b^{5} x^{10} + 10 \, a^{3} b^{4} x^{8} + 5 \, a^{4} b^{3} x^{6} + a^{5} b^{2} x^{4}\right )} \log \left (x\right )}{40 \,{\left (a^{8} b^{5} x^{14} + 5 \, a^{9} b^{4} x^{12} + 10 \, a^{10} b^{3} x^{10} + 10 \, a^{11} b^{2} x^{8} + 5 \, a^{12} b x^{6} + a^{13} x^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^4 + 2*a*b*x^2 + a^2)^3*x^5),x, algorithm="fricas")

[Out]

1/40*(420*a*b^6*x^12 + 1890*a^2*b^5*x^10 + 3290*a^3*b^4*x^8 + 2695*a^4*b^3*x^6 +
959*a^5*b^2*x^4 + 70*a^6*b*x^2 - 10*a^7 - 420*(b^7*x^14 + 5*a*b^6*x^12 + 10*a^2
*b^5*x^10 + 10*a^3*b^4*x^8 + 5*a^4*b^3*x^6 + a^5*b^2*x^4)*log(b*x^2 + a) + 840*(
b^7*x^14 + 5*a*b^6*x^12 + 10*a^2*b^5*x^10 + 10*a^3*b^4*x^8 + 5*a^4*b^3*x^6 + a^5
*b^2*x^4)*log(x))/(a^8*b^5*x^14 + 5*a^9*b^4*x^12 + 10*a^10*b^3*x^10 + 10*a^11*b^
2*x^8 + 5*a^12*b*x^6 + a^13*x^4)

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Sympy [A]  time = 59.9091, size = 165, normalized size = 1.18 \[ \frac{- 10 a^{6} + 70 a^{5} b x^{2} + 959 a^{4} b^{2} x^{4} + 2695 a^{3} b^{3} x^{6} + 3290 a^{2} b^{4} x^{8} + 1890 a b^{5} x^{10} + 420 b^{6} x^{12}}{40 a^{12} x^{4} + 200 a^{11} b x^{6} + 400 a^{10} b^{2} x^{8} + 400 a^{9} b^{3} x^{10} + 200 a^{8} b^{4} x^{12} + 40 a^{7} b^{5} x^{14}} + \frac{21 b^{2} \log{\left (x \right )}}{a^{8}} - \frac{21 b^{2} \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{8}} \]

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)**3,x)

[Out]

(-10*a**6 + 70*a**5*b*x**2 + 959*a**4*b**2*x**4 + 2695*a**3*b**3*x**6 + 3290*a**
2*b**4*x**8 + 1890*a*b**5*x**10 + 420*b**6*x**12)/(40*a**12*x**4 + 200*a**11*b*x
**6 + 400*a**10*b**2*x**8 + 400*a**9*b**3*x**10 + 200*a**8*b**4*x**12 + 40*a**7*
b**5*x**14) + 21*b**2*log(x)/a**8 - 21*b**2*log(a/b + x**2)/(2*a**8)

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GIAC/XCAS [A]  time = 0.272207, size = 176, normalized size = 1.26 \[ \frac{21 \, b^{2}{\rm ln}\left (x^{2}\right )}{2 \, a^{8}} - \frac{21 \, b^{2}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{8}} - \frac{63 \, b^{2} x^{4} - 12 \, a b x^{2} + a^{2}}{4 \, a^{8} x^{4}} + \frac{959 \, b^{7} x^{10} + 5095 \, a b^{6} x^{8} + 10890 \, a^{2} b^{5} x^{6} + 11730 \, a^{3} b^{4} x^{4} + 6390 \, a^{4} b^{3} x^{2} + 1418 \, a^{5} b^{2}}{40 \,{\left (b x^{2} + a\right )}^{5} a^{8}} \]

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^5),x, algorithm="giac")

[Out]

21/2*b^2*ln(x^2)/a^8 - 21/2*b^2*ln(abs(b*x^2 + a))/a^8 - 1/4*(63*b^2*x^4 - 12*a*
b*x^2 + a^2)/(a^8*x^4) + 1/40*(959*b^7*x^10 + 5095*a*b^6*x^8 + 10890*a^2*b^5*x^6
+ 11730*a^3*b^4*x^4 + 6390*a^4*b^3*x^2 + 1418*a^5*b^2)/((b*x^2 + a)^5*a^8)