3.1643 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^3 x^8} \, dx\)

Optimal. Leaf size=97 \[ \frac{15 a^4 \log (x)}{b^7}-\frac{15 a^4 \log (a x+b)}{b^7}+\frac{5 a^4}{b^6 (a x+b)}+\frac{a^4}{2 b^5 (a x+b)^2}+\frac{10 a^3}{b^6 x}-\frac{3 a^2}{b^5 x^2}+\frac{a}{b^4 x^3}-\frac{1}{4 b^3 x^4} \]

[Out]

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

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Rubi [A]  time = 0.148779, antiderivative size = 97, 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 \[ \frac{15 a^4 \log (x)}{b^7}-\frac{15 a^4 \log (a x+b)}{b^7}+\frac{5 a^4}{b^6 (a x+b)}+\frac{a^4}{2 b^5 (a x+b)^2}+\frac{10 a^3}{b^6 x}-\frac{3 a^2}{b^5 x^2}+\frac{a}{b^4 x^3}-\frac{1}{4 b^3 x^4} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^3*x^8),x]

[Out]

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

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Rubi in Sympy [A]  time = 22.1856, size = 95, normalized size = 0.98 \[ \frac{a^{4}}{2 b^{5} \left (a x + b\right )^{2}} + \frac{5 a^{4}}{b^{6} \left (a x + b\right )} + \frac{15 a^{4} \log{\left (x \right )}}{b^{7}} - \frac{15 a^{4} \log{\left (a x + b \right )}}{b^{7}} + \frac{10 a^{3}}{b^{6} x} - \frac{3 a^{2}}{b^{5} x^{2}} + \frac{a}{b^{4} x^{3}} - \frac{1}{4 b^{3} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x)**3/x**8,x)

[Out]

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

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Mathematica [A]  time = 0.104404, size = 90, normalized size = 0.93 \[ \frac{-60 a^4 \log (a x+b)+60 a^4 \log (x)+\frac{b \left (60 a^5 x^5+90 a^4 b x^4+20 a^3 b^2 x^3-5 a^2 b^3 x^2+2 a b^4 x-b^5\right )}{x^4 (a x+b)^2}}{4 b^7} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x)^3*x^8),x]

[Out]

((b*(-b^5 + 2*a*b^4*x - 5*a^2*b^3*x^2 + 20*a^3*b^2*x^3 + 90*a^4*b*x^4 + 60*a^5*x
^5))/(x^4*(b + a*x)^2) + 60*a^4*Log[x] - 60*a^4*Log[b + a*x])/(4*b^7)

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Maple [A]  time = 0.018, size = 94, normalized size = 1. \[ -{\frac{1}{4\,{b}^{3}{x}^{4}}}+{\frac{a}{{b}^{4}{x}^{3}}}-3\,{\frac{{a}^{2}}{{b}^{5}{x}^{2}}}+10\,{\frac{{a}^{3}}{{b}^{6}x}}+{\frac{{a}^{4}}{2\,{b}^{5} \left ( ax+b \right ) ^{2}}}+5\,{\frac{{a}^{4}}{{b}^{6} \left ( ax+b \right ) }}+15\,{\frac{{a}^{4}\ln \left ( x \right ) }{{b}^{7}}}-15\,{\frac{{a}^{4}\ln \left ( ax+b \right ) }{{b}^{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^3/x^8,x)

[Out]

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

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Maxima [A]  time = 1.44143, size = 146, normalized size = 1.51 \[ \frac{60 \, a^{5} x^{5} + 90 \, a^{4} b x^{4} + 20 \, a^{3} b^{2} x^{3} - 5 \, a^{2} b^{3} x^{2} + 2 \, a b^{4} x - b^{5}}{4 \,{\left (a^{2} b^{6} x^{6} + 2 \, a b^{7} x^{5} + b^{8} x^{4}\right )}} - \frac{15 \, a^{4} \log \left (a x + b\right )}{b^{7}} + \frac{15 \, a^{4} \log \left (x\right )}{b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^3*x^8),x, algorithm="maxima")

[Out]

1/4*(60*a^5*x^5 + 90*a^4*b*x^4 + 20*a^3*b^2*x^3 - 5*a^2*b^3*x^2 + 2*a*b^4*x - b^
5)/(a^2*b^6*x^6 + 2*a*b^7*x^5 + b^8*x^4) - 15*a^4*log(a*x + b)/b^7 + 15*a^4*log(
x)/b^7

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Fricas [A]  time = 0.230334, size = 205, normalized size = 2.11 \[ \frac{60 \, a^{5} b x^{5} + 90 \, a^{4} b^{2} x^{4} + 20 \, a^{3} b^{3} x^{3} - 5 \, a^{2} b^{4} x^{2} + 2 \, a b^{5} x - b^{6} - 60 \,{\left (a^{6} x^{6} + 2 \, a^{5} b x^{5} + a^{4} b^{2} x^{4}\right )} \log \left (a x + b\right ) + 60 \,{\left (a^{6} x^{6} + 2 \, a^{5} b x^{5} + a^{4} b^{2} x^{4}\right )} \log \left (x\right )}{4 \,{\left (a^{2} b^{7} x^{6} + 2 \, a b^{8} x^{5} + b^{9} x^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^3*x^8),x, algorithm="fricas")

[Out]

1/4*(60*a^5*b*x^5 + 90*a^4*b^2*x^4 + 20*a^3*b^3*x^3 - 5*a^2*b^4*x^2 + 2*a*b^5*x
- b^6 - 60*(a^6*x^6 + 2*a^5*b*x^5 + a^4*b^2*x^4)*log(a*x + b) + 60*(a^6*x^6 + 2*
a^5*b*x^5 + a^4*b^2*x^4)*log(x))/(a^2*b^7*x^6 + 2*a*b^8*x^5 + b^9*x^4)

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Sympy [A]  time = 2.44337, size = 102, normalized size = 1.05 \[ \frac{15 a^{4} \left (\log{\left (x \right )} - \log{\left (x + \frac{b}{a} \right )}\right )}{b^{7}} + \frac{60 a^{5} x^{5} + 90 a^{4} b x^{4} + 20 a^{3} b^{2} x^{3} - 5 a^{2} b^{3} x^{2} + 2 a b^{4} x - b^{5}}{4 a^{2} b^{6} x^{6} + 8 a b^{7} x^{5} + 4 b^{8} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x)**3/x**8,x)

[Out]

15*a**4*(log(x) - log(x + b/a))/b**7 + (60*a**5*x**5 + 90*a**4*b*x**4 + 20*a**3*
b**2*x**3 - 5*a**2*b**3*x**2 + 2*a*b**4*x - b**5)/(4*a**2*b**6*x**6 + 8*a*b**7*x
**5 + 4*b**8*x**4)

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GIAC/XCAS [A]  time = 0.228772, size = 131, normalized size = 1.35 \[ -\frac{15 \, a^{4}{\rm ln}\left ({\left | a x + b \right |}\right )}{b^{7}} + \frac{15 \, a^{4}{\rm ln}\left ({\left | x \right |}\right )}{b^{7}} + \frac{60 \, a^{5} b x^{5} + 90 \, a^{4} b^{2} x^{4} + 20 \, a^{3} b^{3} x^{3} - 5 \, a^{2} b^{4} x^{2} + 2 \, a b^{5} x - b^{6}}{4 \,{\left (a x + b\right )}^{2} b^{7} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^3*x^8),x, algorithm="giac")

[Out]

-15*a^4*ln(abs(a*x + b))/b^7 + 15*a^4*ln(abs(x))/b^7 + 1/4*(60*a^5*b*x^5 + 90*a^
4*b^2*x^4 + 20*a^3*b^3*x^3 - 5*a^2*b^4*x^2 + 2*a*b^5*x - b^6)/((a*x + b)^2*b^7*x
^4)