3.1629 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^2 x^7} \, dx\)

Optimal. Leaf size=84 \[ \frac{5 a^4 \log (x)}{b^6}-\frac{5 a^4 \log (a x+b)}{b^6}+\frac{a^4}{b^5 (a x+b)}+\frac{4 a^3}{b^5 x}-\frac{3 a^2}{2 b^4 x^2}+\frac{2 a}{3 b^3 x^3}-\frac{1}{4 b^2 x^4} \]

[Out]

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

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Rubi [A]  time = 0.118348, antiderivative size = 84, 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 \[ \frac{5 a^4 \log (x)}{b^6}-\frac{5 a^4 \log (a x+b)}{b^6}+\frac{a^4}{b^5 (a x+b)}+\frac{4 a^3}{b^5 x}-\frac{3 a^2}{2 b^4 x^2}+\frac{2 a}{3 b^3 x^3}-\frac{1}{4 b^2 x^4} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^2*x^7),x]

[Out]

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

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Rubi in Sympy [A]  time = 17.4817, size = 83, normalized size = 0.99 \[ \frac{a^{4}}{b^{5} \left (a x + b\right )} + \frac{5 a^{4} \log{\left (x \right )}}{b^{6}} - \frac{5 a^{4} \log{\left (a x + b \right )}}{b^{6}} + \frac{4 a^{3}}{b^{5} x} - \frac{3 a^{2}}{2 b^{4} x^{2}} + \frac{2 a}{3 b^{3} x^{3}} - \frac{1}{4 b^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x)**2/x**7,x)

[Out]

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

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Mathematica [A]  time = 0.0819086, size = 79, normalized size = 0.94 \[ \frac{-60 a^4 \log (a x+b)+60 a^4 \log (x)+\frac{b \left (60 a^4 x^4+30 a^3 b x^3-10 a^2 b^2 x^2+5 a b^3 x-3 b^4\right )}{x^4 (a x+b)}}{12 b^6} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x)^2*x^7),x]

[Out]

((b*(-3*b^4 + 5*a*b^3*x - 10*a^2*b^2*x^2 + 30*a^3*b*x^3 + 60*a^4*x^4))/(x^4*(b +
a*x)) + 60*a^4*Log[x] - 60*a^4*Log[b + a*x])/(12*b^6)

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Maple [A]  time = 0.017, size = 79, normalized size = 0.9 \[ -{\frac{1}{4\,{b}^{2}{x}^{4}}}+{\frac{2\,a}{3\,{b}^{3}{x}^{3}}}-{\frac{3\,{a}^{2}}{2\,{b}^{4}{x}^{2}}}+4\,{\frac{{a}^{3}}{{b}^{5}x}}+{\frac{{a}^{4}}{{b}^{5} \left ( ax+b \right ) }}+5\,{\frac{{a}^{4}\ln \left ( x \right ) }{{b}^{6}}}-5\,{\frac{{a}^{4}\ln \left ( ax+b \right ) }{{b}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^2/x^7,x)

[Out]

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

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Maxima [A]  time = 1.42611, size = 116, normalized size = 1.38 \[ \frac{60 \, a^{4} x^{4} + 30 \, a^{3} b x^{3} - 10 \, a^{2} b^{2} x^{2} + 5 \, a b^{3} x - 3 \, b^{4}}{12 \,{\left (a b^{5} x^{5} + b^{6} x^{4}\right )}} - \frac{5 \, a^{4} \log \left (a x + b\right )}{b^{6}} + \frac{5 \, a^{4} \log \left (x\right )}{b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^2*x^7),x, algorithm="maxima")

[Out]

1/12*(60*a^4*x^4 + 30*a^3*b*x^3 - 10*a^2*b^2*x^2 + 5*a*b^3*x - 3*b^4)/(a*b^5*x^5
+ b^6*x^4) - 5*a^4*log(a*x + b)/b^6 + 5*a^4*log(x)/b^6

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Fricas [A]  time = 0.233272, size = 146, normalized size = 1.74 \[ \frac{60 \, a^{4} b x^{4} + 30 \, a^{3} b^{2} x^{3} - 10 \, a^{2} b^{3} x^{2} + 5 \, a b^{4} x - 3 \, b^{5} - 60 \,{\left (a^{5} x^{5} + a^{4} b x^{4}\right )} \log \left (a x + b\right ) + 60 \,{\left (a^{5} x^{5} + a^{4} b x^{4}\right )} \log \left (x\right )}{12 \,{\left (a b^{6} x^{5} + b^{7} x^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^2*x^7),x, algorithm="fricas")

[Out]

1/12*(60*a^4*b*x^4 + 30*a^3*b^2*x^3 - 10*a^2*b^3*x^2 + 5*a*b^4*x - 3*b^5 - 60*(a
^5*x^5 + a^4*b*x^4)*log(a*x + b) + 60*(a^5*x^5 + a^4*b*x^4)*log(x))/(a*b^6*x^5 +
b^7*x^4)

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Sympy [A]  time = 2.02552, size = 80, normalized size = 0.95 \[ \frac{5 a^{4} \left (\log{\left (x \right )} - \log{\left (x + \frac{b}{a} \right )}\right )}{b^{6}} + \frac{60 a^{4} x^{4} + 30 a^{3} b x^{3} - 10 a^{2} b^{2} x^{2} + 5 a b^{3} x - 3 b^{4}}{12 a b^{5} x^{5} + 12 b^{6} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x)**2/x**7,x)

[Out]

5*a**4*(log(x) - log(x + b/a))/b**6 + (60*a**4*x**4 + 30*a**3*b*x**3 - 10*a**2*b
**2*x**2 + 5*a*b**3*x - 3*b**4)/(12*a*b**5*x**5 + 12*b**6*x**4)

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GIAC/XCAS [A]  time = 0.224528, size = 116, normalized size = 1.38 \[ -\frac{5 \, a^{4}{\rm ln}\left ({\left | a x + b \right |}\right )}{b^{6}} + \frac{5 \, a^{4}{\rm ln}\left ({\left | x \right |}\right )}{b^{6}} + \frac{60 \, a^{4} b x^{4} + 30 \, a^{3} b^{2} x^{3} - 10 \, a^{2} b^{3} x^{2} + 5 \, a b^{4} x - 3 \, b^{5}}{12 \,{\left (a x + b\right )} b^{6} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^2*x^7),x, algorithm="giac")

[Out]

-5*a^4*ln(abs(a*x + b))/b^6 + 5*a^4*ln(abs(x))/b^6 + 1/12*(60*a^4*b*x^4 + 30*a^3
*b^2*x^3 - 10*a^2*b^3*x^2 + 5*a*b^4*x - 3*b^5)/((a*x + b)*b^6*x^4)