3.1630 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^2 x^8} \, dx\)

Optimal. Leaf size=94 \[ -\frac{6 a^5 \log (x)}{b^7}+\frac{6 a^5 \log (a x+b)}{b^7}-\frac{a^5}{b^6 (a x+b)}-\frac{5 a^4}{b^6 x}+\frac{2 a^3}{b^5 x^2}-\frac{a^2}{b^4 x^3}+\frac{a}{2 b^3 x^4}-\frac{1}{5 b^2 x^5} \]

[Out]

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

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Rubi [A]  time = 0.136887, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{6 a^5 \log (x)}{b^7}+\frac{6 a^5 \log (a x+b)}{b^7}-\frac{a^5}{b^6 (a x+b)}-\frac{5 a^4}{b^6 x}+\frac{2 a^3}{b^5 x^2}-\frac{a^2}{b^4 x^3}+\frac{a}{2 b^3 x^4}-\frac{1}{5 b^2 x^5} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^2*x^8),x]

[Out]

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

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Rubi in Sympy [A]  time = 19.3307, size = 90, normalized size = 0.96 \[ - \frac{a^{5}}{b^{6} \left (a x + b\right )} - \frac{6 a^{5} \log{\left (x \right )}}{b^{7}} + \frac{6 a^{5} \log{\left (a x + b \right )}}{b^{7}} - \frac{5 a^{4}}{b^{6} x} + \frac{2 a^{3}}{b^{5} x^{2}} - \frac{a^{2}}{b^{4} x^{3}} + \frac{a}{2 b^{3} x^{4}} - \frac{1}{5 b^{2} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x)**2/x**8,x)

[Out]

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

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Mathematica [A]  time = 0.132826, size = 90, normalized size = 0.96 \[ -\frac{-60 a^5 \log (a x+b)+60 a^5 \log (x)+\frac{b \left (60 a^5 x^5+30 a^4 b x^4-10 a^3 b^2 x^3+5 a^2 b^3 x^2-3 a b^4 x+2 b^5\right )}{x^5 (a x+b)}}{10 b^7} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x)^2*x^8),x]

[Out]

-((b*(2*b^5 - 3*a*b^4*x + 5*a^2*b^3*x^2 - 10*a^3*b^2*x^3 + 30*a^4*b*x^4 + 60*a^5
*x^5))/(x^5*(b + a*x)) + 60*a^5*Log[x] - 60*a^5*Log[b + a*x])/(10*b^7)

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Maple [A]  time = 0.017, size = 91, normalized size = 1. \[ -{\frac{1}{5\,{b}^{2}{x}^{5}}}+{\frac{a}{2\,{b}^{3}{x}^{4}}}-{\frac{{a}^{2}}{{b}^{4}{x}^{3}}}+2\,{\frac{{a}^{3}}{{b}^{5}{x}^{2}}}-5\,{\frac{{a}^{4}}{{b}^{6}x}}-{\frac{{a}^{5}}{{b}^{6} \left ( ax+b \right ) }}-6\,{\frac{{a}^{5}\ln \left ( x \right ) }{{b}^{7}}}+6\,{\frac{{a}^{5}\ln \left ( ax+b \right ) }{{b}^{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^2/x^8,x)

[Out]

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

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Maxima [A]  time = 1.45136, size = 131, normalized size = 1.39 \[ -\frac{60 \, a^{5} x^{5} + 30 \, a^{4} b x^{4} - 10 \, a^{3} b^{2} x^{3} + 5 \, a^{2} b^{3} x^{2} - 3 \, a b^{4} x + 2 \, b^{5}}{10 \,{\left (a b^{6} x^{6} + b^{7} x^{5}\right )}} + \frac{6 \, a^{5} \log \left (a x + b\right )}{b^{7}} - \frac{6 \, a^{5} \log \left (x\right )}{b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^2*x^8),x, algorithm="maxima")

[Out]

-1/10*(60*a^5*x^5 + 30*a^4*b*x^4 - 10*a^3*b^2*x^3 + 5*a^2*b^3*x^2 - 3*a*b^4*x +
2*b^5)/(a*b^6*x^6 + b^7*x^5) + 6*a^5*log(a*x + b)/b^7 - 6*a^5*log(x)/b^7

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Fricas [A]  time = 0.225915, size = 161, normalized size = 1.71 \[ -\frac{60 \, a^{5} b x^{5} + 30 \, a^{4} b^{2} x^{4} - 10 \, a^{3} b^{3} x^{3} + 5 \, a^{2} b^{4} x^{2} - 3 \, a b^{5} x + 2 \, b^{6} - 60 \,{\left (a^{6} x^{6} + a^{5} b x^{5}\right )} \log \left (a x + b\right ) + 60 \,{\left (a^{6} x^{6} + a^{5} b x^{5}\right )} \log \left (x\right )}{10 \,{\left (a b^{7} x^{6} + b^{8} x^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^2*x^8),x, algorithm="fricas")

[Out]

-1/10*(60*a^5*b*x^5 + 30*a^4*b^2*x^4 - 10*a^3*b^3*x^3 + 5*a^2*b^4*x^2 - 3*a*b^5*
x + 2*b^6 - 60*(a^6*x^6 + a^5*b*x^5)*log(a*x + b) + 60*(a^6*x^6 + a^5*b*x^5)*log
(x))/(a*b^7*x^6 + b^8*x^5)

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Sympy [A]  time = 2.22792, size = 92, normalized size = 0.98 \[ \frac{6 a^{5} \left (- \log{\left (x \right )} + \log{\left (x + \frac{b}{a} \right )}\right )}{b^{7}} - \frac{60 a^{5} x^{5} + 30 a^{4} b x^{4} - 10 a^{3} b^{2} x^{3} + 5 a^{2} b^{3} x^{2} - 3 a b^{4} x + 2 b^{5}}{10 a b^{6} x^{6} + 10 b^{7} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x)**2/x**8,x)

[Out]

6*a**5*(-log(x) + log(x + b/a))/b**7 - (60*a**5*x**5 + 30*a**4*b*x**4 - 10*a**3*
b**2*x**3 + 5*a**2*b**3*x**2 - 3*a*b**4*x + 2*b**5)/(10*a*b**6*x**6 + 10*b**7*x*
*5)

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GIAC/XCAS [A]  time = 0.229164, size = 131, normalized size = 1.39 \[ \frac{6 \, a^{5}{\rm ln}\left ({\left | a x + b \right |}\right )}{b^{7}} - \frac{6 \, a^{5}{\rm ln}\left ({\left | x \right |}\right )}{b^{7}} - \frac{60 \, a^{5} b x^{5} + 30 \, a^{4} b^{2} x^{4} - 10 \, a^{3} b^{3} x^{3} + 5 \, a^{2} b^{4} x^{2} - 3 \, a b^{5} x + 2 \, b^{6}}{10 \,{\left (a x + b\right )} b^{7} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^2*x^8),x, algorithm="giac")

[Out]

6*a^5*ln(abs(a*x + b))/b^7 - 6*a^5*ln(abs(x))/b^7 - 1/10*(60*a^5*b*x^5 + 30*a^4*
b^2*x^4 - 10*a^3*b^3*x^3 + 5*a^2*b^4*x^2 - 3*a*b^5*x + 2*b^6)/((a*x + b)*b^7*x^5
)