3.1618 \(\int \frac {1}{(a+\frac {b}{x}) x^7} \, dx\)

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+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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Rubi [A]  time = 0.03, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {263, 44} \[ \frac {a^3}{2 b^4 x^2}-\frac {a^2}{3 b^3 x^3}-\frac {a^4}{b^5 x}-\frac {a^5 \log (x)}{b^6}+\frac {a^5 \log (a x+b)}{b^6}+\frac {a}{4 b^2 x^4}-\frac {1}{5 b x^5} \]

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

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rubi steps

\begin {align*} \int \frac {1}{\left (a+\frac {b}{x}\right ) x^7} \, dx &=\int \frac {1}{x^6 (b+a x)} \, dx\\ &=\int \left (\frac {1}{b x^6}-\frac {a}{b^2 x^5}+\frac {a^2}{b^3 x^4}-\frac {a^3}{b^4 x^3}+\frac {a^4}{b^5 x^2}-\frac {a^5}{b^6 x}+\frac {a^6}{b^6 (b+a x)}\right ) \, dx\\ &=-\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 \log (x)}{b^6}+\frac {a^5 \log (b+a x)}{b^6}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 82, normalized size = 1.00 \[ -\frac {a^5 \log (x)}{b^6}+\frac {a^5 \log (a x+b)}{b^6}-\frac {a^4}{b^5 x}+\frac {a^3}{2 b^4 x^2}-\frac {a^2}{3 b^3 x^3}+\frac {a}{4 b^2 x^4}-\frac {1}{5 b x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x)*x^7),x]

[Out]

-1/5*1/(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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fricas [A]  time = 1.13, size = 76, normalized size = 0.93 \[ \frac {60 \, a^{5} x^{5} \log \left (a x + b\right ) - 60 \, a^{5} x^{5} \log \relax (x) - 60 \, a^{4} b x^{4} + 30 \, a^{3} b^{2} x^{3} - 20 \, a^{2} b^{3} x^{2} + 15 \, a b^{4} x - 12 \, b^{5}}{60 \, b^{6} x^{5}} \]

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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giac [A]  time = 0.17, size = 78, normalized size = 0.95 \[ \frac {a^{5} \log \left ({\left | a x + b \right |}\right )}{b^{6}} - \frac {a^{5} \log \left ({\left | x \right |}\right )}{b^{6}} - \frac {60 \, a^{4} b x^{4} - 30 \, a^{3} b^{2} x^{3} + 20 \, a^{2} b^{3} x^{2} - 15 \, a b^{4} x + 12 \, b^{5}}{60 \, b^{6} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^5*log(abs(a*x + b))/b^6 - a^5*log(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)

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maple [A]  time = 0.01, size = 75, normalized size = 0.91 \[ -\frac {a^{5} \ln \relax (x )}{b^{6}}+\frac {a^{5} \ln \left (a x +b \right )}{b^{6}}-\frac {a^{4}}{b^{5} x}+\frac {a^{3}}{2 b^{4} x^{2}}-\frac {a^{2}}{3 b^{3} x^{3}}+\frac {a}{4 b^{2} x^{4}}-\frac {1}{5 b \,x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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 = 1.07, size = 73, normalized size = 0.89 \[ \frac {a^{5} \log \left (a x + b\right )}{b^{6}} - \frac {a^{5} \log \relax (x)}{b^{6}} - \frac {60 \, a^{4} x^{4} - 30 \, a^{3} b x^{3} + 20 \, a^{2} b^{2} x^{2} - 15 \, a b^{3} x + 12 \, b^{4}}{60 \, b^{5} x^{5}} \]

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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mupad [B]  time = 1.10, size = 70, normalized size = 0.85 \[ \frac {2\,a^5\,\mathrm {atanh}\left (\frac {2\,a\,x}{b}+1\right )}{b^6}-\frac {a^4\,b\,x^4-\frac {a^3\,b^2\,x^3}{2}+\frac {a^2\,b^3\,x^2}{3}-\frac {a\,b^4\,x}{4}+\frac {b^5}{5}}{b^6\,x^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^7*(a + b/x)),x)

[Out]

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

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sympy [A]  time = 0.30, size = 68, normalized size = 0.83 \[ \frac {a^{5} \left (- \log {\relax (x )} + \log {\left (x + \frac {b}{a} \right )}\right )}{b^{6}} + \frac {- 60 a^{4} x^{4} + 30 a^{3} b x^{3} - 20 a^{2} b^{2} x^{2} + 15 a b^{3} x - 12 b^{4}}{60 b^{5} x^{5}} \]

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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