3.1632 \(\int \frac {1}{(a+\frac {b}{x})^2 x^8} \, dx\)

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+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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Rubi [A]  time = 0.05, antiderivative size = 94, 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} \[ \frac {2 a^3}{b^5 x^2}-\frac {a^2}{b^4 x^3}-\frac {a^5}{b^6 (a x+b)}-\frac {5 a^4}{b^6 x}-\frac {6 a^5 \log (x)}{b^7}+\frac {6 a^5 \log (a x+b)}{b^7}+\frac {a}{2 b^3 x^4}-\frac {1}{5 b^2 x^5} \]

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

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 )^2 x^8} \, dx &=\int \frac {1}{x^6 (b+a x)^2} \, dx\\ &=\int \left (\frac {1}{b^2 x^6}-\frac {2 a}{b^3 x^5}+\frac {3 a^2}{b^4 x^4}-\frac {4 a^3}{b^5 x^3}+\frac {5 a^4}{b^6 x^2}-\frac {6 a^5}{b^7 x}+\frac {a^6}{b^6 (b+a x)^2}+\frac {6 a^6}{b^7 (b+a x)}\right ) \, dx\\ &=-\frac {1}{5 b^2 x^5}+\frac {a}{2 b^3 x^4}-\frac {a^2}{b^4 x^3}+\frac {2 a^3}{b^5 x^2}-\frac {5 a^4}{b^6 x}-\frac {a^5}{b^6 (b+a x)}-\frac {6 a^5 \log (x)}{b^7}+\frac {6 a^5 \log (b+a x)}{b^7}\\ \end {align*}

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Mathematica [A]  time = 0.07, 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]

-1/10*((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])/b^7

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fricas [A]  time = 0.98, size = 119, normalized size = 1.27 \[ -\frac {60 \, a^{5} b x^{5} + 30 \, a^{4} b^{2} x^{4} - 10 \, a^{3} b^{3} x^{3} + 5 \, a^{2} b^{4} x^{2} - 3 \, a b^{5} x + 2 \, b^{6} - 60 \, {\left (a^{6} x^{6} + a^{5} b x^{5}\right )} \log \left (a x + b\right ) + 60 \, {\left (a^{6} x^{6} + a^{5} b x^{5}\right )} \log \relax (x)}{10 \, {\left (a b^{7} x^{6} + b^{8} x^{5}\right )}} \]

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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giac [A]  time = 0.15, size = 97, normalized size = 1.03 \[ \frac {6 \, a^{5} \log \left ({\left | a x + b \right |}\right )}{b^{7}} - \frac {6 \, a^{5} \log \left ({\left | x \right |}\right )}{b^{7}} - \frac {60 \, a^{5} b x^{5} + 30 \, a^{4} b^{2} x^{4} - 10 \, a^{3} b^{3} x^{3} + 5 \, a^{2} b^{4} x^{2} - 3 \, a b^{5} x + 2 \, b^{6}}{10 \, {\left (a x + b\right )} b^{7} x^{5}} \]

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*log(abs(a*x + b))/b^7 - 6*a^5*log(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)

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maple [A]  time = 0.01, size = 91, normalized size = 0.97 \[ -\frac {a^{5}}{\left (a x +b \right ) b^{6}}-\frac {6 a^{5} \ln \relax (x )}{b^{7}}+\frac {6 a^{5} \ln \left (a x +b \right )}{b^{7}}-\frac {5 a^{4}}{b^{6} x}+\frac {2 a^{3}}{b^{5} x^{2}}-\frac {a^{2}}{b^{4} x^{3}}+\frac {a}{2 b^{3} x^{4}}-\frac {1}{5 b^{2} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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 = 1.09, size = 97, normalized size = 1.03 \[ -\frac {60 \, a^{5} x^{5} + 30 \, a^{4} b x^{4} - 10 \, a^{3} b^{2} x^{3} + 5 \, a^{2} b^{3} x^{2} - 3 \, a b^{4} x + 2 \, b^{5}}{10 \, {\left (a b^{6} x^{6} + b^{7} x^{5}\right )}} + \frac {6 \, a^{5} \log \left (a x + b\right )}{b^{7}} - \frac {6 \, a^{5} \log \relax (x)}{b^{7}} \]

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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mupad [B]  time = 1.10, size = 91, normalized size = 0.97 \[ \frac {12\,a^5\,\mathrm {atanh}\left (\frac {2\,a\,x}{b}+1\right )}{b^7}-\frac {\frac {1}{5\,b}+\frac {a^2\,x^2}{2\,b^3}-\frac {a^3\,x^3}{b^4}+\frac {3\,a^4\,x^4}{b^5}+\frac {6\,a^5\,x^5}{b^6}-\frac {3\,a\,x}{10\,b^2}}{a\,x^6+b\,x^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^8*(a + b/x)^2),x)

[Out]

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

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sympy [A]  time = 0.43, size = 92, normalized size = 0.98 \[ \frac {6 a^{5} \left (- \log {\relax (x )} + \log {\left (x + \frac {b}{a} \right )}\right )}{b^{7}} + \frac {- 60 a^{5} x^{5} - 30 a^{4} b x^{4} + 10 a^{3} b^{2} x^{3} - 5 a^{2} b^{3} x^{2} + 3 a b^{4} x - 2 b^{5}}{10 a b^{6} x^{6} + 10 b^{7} x^{5}} \]

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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