3.1616 \(\int \frac{1}{\left (a+\frac{b}{x}\right ) x^7} \, dx\)

Optimal. Leaf size=82 \[ -\frac{a^5 \log (x)}{b^6}+\frac{a^5 \log (a x+b)}{b^6}-\frac{a^4}{b^5 x}+\frac{a^3}{2 b^4 x^2}-\frac{a^2}{3 b^3 x^3}+\frac{a}{4 b^2 x^4}-\frac{1}{5 b x^5} \]

[Out]

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

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Rubi [A]  time = 0.101517, antiderivative size = 82, 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 \[ -\frac{a^5 \log (x)}{b^6}+\frac{a^5 \log (a x+b)}{b^6}-\frac{a^4}{b^5 x}+\frac{a^3}{2 b^4 x^2}-\frac{a^2}{3 b^3 x^3}+\frac{a}{4 b^2 x^4}-\frac{1}{5 b x^5} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)*x^7),x]

[Out]

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

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Rubi in Sympy [A]  time = 14.7726, size = 73, normalized size = 0.89 \[ - \frac{a^{5} \log{\left (x \right )}}{b^{6}} + \frac{a^{5} \log{\left (a x + b \right )}}{b^{6}} - \frac{a^{4}}{b^{5} x} + \frac{a^{3}}{2 b^{4} x^{2}} - \frac{a^{2}}{3 b^{3} x^{3}} + \frac{a}{4 b^{2} x^{4}} - \frac{1}{5 b x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00927439, size = 82, normalized size = 1. \[ -\frac{a^5 \log (x)}{b^6}+\frac{a^5 \log (a x+b)}{b^6}-\frac{a^4}{b^5 x}+\frac{a^3}{2 b^4 x^2}-\frac{a^2}{3 b^3 x^3}+\frac{a}{4 b^2 x^4}-\frac{1}{5 b x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x)*x^7),x]

[Out]

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

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Maple [A]  time = 0.013, size = 75, normalized size = 0.9 \[ -{\frac{1}{5\,b{x}^{5}}}+{\frac{a}{4\,{b}^{2}{x}^{4}}}-{\frac{{a}^{2}}{3\,{b}^{3}{x}^{3}}}+{\frac{{a}^{3}}{2\,{b}^{4}{x}^{2}}}-{\frac{{a}^{4}}{{b}^{5}x}}-{\frac{{a}^{5}\ln \left ( x \right ) }{{b}^{6}}}+{\frac{{a}^{5}\ln \left ( ax+b \right ) }{{b}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)/x^7,x)

[Out]

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

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Maxima [A]  time = 1.44793, size = 99, normalized size = 1.21 \[ \frac{a^{5} \log \left (a x + b\right )}{b^{6}} - \frac{a^{5} \log \left (x\right )}{b^{6}} - \frac{60 \, a^{4} x^{4} - 30 \, a^{3} b x^{3} + 20 \, a^{2} b^{2} x^{2} - 15 \, a b^{3} x + 12 \, b^{4}}{60 \, b^{5} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a^5*log(a*x + b)/b^6 - a^5*log(x)/b^6 - 1/60*(60*a^4*x^4 - 30*a^3*b*x^3 + 20*a^2
*b^2*x^2 - 15*a*b^3*x + 12*b^4)/(b^5*x^5)

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Fricas [A]  time = 0.226477, size = 103, normalized size = 1.26 \[ \frac{60 \, a^{5} x^{5} \log \left (a x + b\right ) - 60 \, a^{5} x^{5} \log \left (x\right ) - 60 \, a^{4} b x^{4} + 30 \, a^{3} b^{2} x^{3} - 20 \, a^{2} b^{3} x^{2} + 15 \, a b^{4} x - 12 \, b^{5}}{60 \, b^{6} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/60*(60*a^5*x^5*log(a*x + b) - 60*a^5*x^5*log(x) - 60*a^4*b*x^4 + 30*a^3*b^2*x^
3 - 20*a^2*b^3*x^2 + 15*a*b^4*x - 12*b^5)/(b^6*x^5)

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Sympy [A]  time = 1.79143, size = 68, normalized size = 0.83 \[ \frac{a^{5} \left (- \log{\left (x \right )} + \log{\left (x + \frac{b}{a} \right )}\right )}{b^{6}} - \frac{60 a^{4} x^{4} - 30 a^{3} b x^{3} + 20 a^{2} b^{2} x^{2} - 15 a b^{3} x + 12 b^{4}}{60 b^{5} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x)/x**7,x)

[Out]

a**5*(-log(x) + log(x + b/a))/b**6 - (60*a**4*x**4 - 30*a**3*b*x**3 + 20*a**2*b*
*2*x**2 - 15*a*b**3*x + 12*b**4)/(60*b**5*x**5)

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GIAC/XCAS [A]  time = 0.222957, size = 105, normalized size = 1.28 \[ \frac{a^{5}{\rm ln}\left ({\left | a x + b \right |}\right )}{b^{6}} - \frac{a^{5}{\rm ln}\left ({\left | x \right |}\right )}{b^{6}} - \frac{60 \, a^{4} b x^{4} - 30 \, a^{3} b^{2} x^{3} + 20 \, a^{2} b^{3} x^{2} - 15 \, a b^{4} x + 12 \, b^{5}}{60 \, b^{6} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a^5*ln(abs(a*x + b))/b^6 - a^5*ln(abs(x))/b^6 - 1/60*(60*a^4*b*x^4 - 30*a^3*b^2*
x^3 + 20*a^2*b^3*x^2 - 15*a*b^4*x + 12*b^5)/(b^6*x^5)