3.1646 \(\int \frac {1}{(a+\frac {b}{x})^3 x^9} \, dx\)

Optimal. Leaf size=111 \[ -\frac {21 a^5 \log (x)}{b^8}+\frac {21 a^5 \log (a x+b)}{b^8}-\frac {6 a^5}{b^7 (a x+b)}-\frac {a^5}{2 b^6 (a x+b)^2}-\frac {15 a^4}{b^7 x}+\frac {5 a^3}{b^6 x^2}-\frac {2 a^2}{b^5 x^3}+\frac {3 a}{4 b^4 x^4}-\frac {1}{5 b^3 x^5} \]

[Out]

-1/5/b^3/x^5+3/4*a/b^4/x^4-2*a^2/b^5/x^3+5*a^3/b^6/x^2-15*a^4/b^7/x-1/2*a^5/b^6/(a*x+b)^2-6*a^5/b^7/(a*x+b)-21
*a^5*ln(x)/b^8+21*a^5*ln(a*x+b)/b^8

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Rubi [A]  time = 0.07, antiderivative size = 111, 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} \[ \frac {5 a^3}{b^6 x^2}-\frac {2 a^2}{b^5 x^3}-\frac {6 a^5}{b^7 (a x+b)}-\frac {a^5}{2 b^6 (a x+b)^2}-\frac {15 a^4}{b^7 x}-\frac {21 a^5 \log (x)}{b^8}+\frac {21 a^5 \log (a x+b)}{b^8}+\frac {3 a}{4 b^4 x^4}-\frac {1}{5 b^3 x^5} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x)^3*x^9),x]

[Out]

-1/(5*b^3*x^5) + (3*a)/(4*b^4*x^4) - (2*a^2)/(b^5*x^3) + (5*a^3)/(b^6*x^2) - (15*a^4)/(b^7*x) - a^5/(2*b^6*(b
+ a*x)^2) - (6*a^5)/(b^7*(b + a*x)) - (21*a^5*Log[x])/b^8 + (21*a^5*Log[b + a*x])/b^8

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 )^3 x^9} \, dx &=\int \frac {1}{x^6 (b+a x)^3} \, dx\\ &=\int \left (\frac {1}{b^3 x^6}-\frac {3 a}{b^4 x^5}+\frac {6 a^2}{b^5 x^4}-\frac {10 a^3}{b^6 x^3}+\frac {15 a^4}{b^7 x^2}-\frac {21 a^5}{b^8 x}+\frac {a^6}{b^6 (b+a x)^3}+\frac {6 a^6}{b^7 (b+a x)^2}+\frac {21 a^6}{b^8 (b+a x)}\right ) \, dx\\ &=-\frac {1}{5 b^3 x^5}+\frac {3 a}{4 b^4 x^4}-\frac {2 a^2}{b^5 x^3}+\frac {5 a^3}{b^6 x^2}-\frac {15 a^4}{b^7 x}-\frac {a^5}{2 b^6 (b+a x)^2}-\frac {6 a^5}{b^7 (b+a x)}-\frac {21 a^5 \log (x)}{b^8}+\frac {21 a^5 \log (b+a x)}{b^8}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 101, normalized size = 0.91 \[ -\frac {-420 a^5 \log (a x+b)+420 a^5 \log (x)+\frac {b \left (420 a^6 x^6+630 a^5 b x^5+140 a^4 b^2 x^4-35 a^3 b^3 x^3+14 a^2 b^4 x^2-7 a b^5 x+4 b^6\right )}{x^5 (a x+b)^2}}{20 b^8} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x)^3*x^9),x]

[Out]

-1/20*((b*(4*b^6 - 7*a*b^5*x + 14*a^2*b^4*x^2 - 35*a^3*b^3*x^3 + 140*a^4*b^2*x^4 + 630*a^5*b*x^5 + 420*a^6*x^6
))/(x^5*(b + a*x)^2) + 420*a^5*Log[x] - 420*a^5*Log[b + a*x])/b^8

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fricas [A]  time = 0.95, size = 163, normalized size = 1.47 \[ -\frac {420 \, a^{6} b x^{6} + 630 \, a^{5} b^{2} x^{5} + 140 \, a^{4} b^{3} x^{4} - 35 \, a^{3} b^{4} x^{3} + 14 \, a^{2} b^{5} x^{2} - 7 \, a b^{6} x + 4 \, b^{7} - 420 \, {\left (a^{7} x^{7} + 2 \, a^{6} b x^{6} + a^{5} b^{2} x^{5}\right )} \log \left (a x + b\right ) + 420 \, {\left (a^{7} x^{7} + 2 \, a^{6} b x^{6} + a^{5} b^{2} x^{5}\right )} \log \relax (x)}{20 \, {\left (a^{2} b^{8} x^{7} + 2 \, a b^{9} x^{6} + b^{10} x^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^3/x^9,x, algorithm="fricas")

[Out]

-1/20*(420*a^6*b*x^6 + 630*a^5*b^2*x^5 + 140*a^4*b^3*x^4 - 35*a^3*b^4*x^3 + 14*a^2*b^5*x^2 - 7*a*b^6*x + 4*b^7
- 420*(a^7*x^7 + 2*a^6*b*x^6 + a^5*b^2*x^5)*log(a*x + b) + 420*(a^7*x^7 + 2*a^6*b*x^6 + a^5*b^2*x^5)*log(x))/
(a^2*b^8*x^7 + 2*a*b^9*x^6 + b^10*x^5)

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giac [A]  time = 0.15, size = 108, normalized size = 0.97 \[ \frac {21 \, a^{5} \log \left ({\left | a x + b \right |}\right )}{b^{8}} - \frac {21 \, a^{5} \log \left ({\left | x \right |}\right )}{b^{8}} - \frac {420 \, a^{6} b x^{6} + 630 \, a^{5} b^{2} x^{5} + 140 \, a^{4} b^{3} x^{4} - 35 \, a^{3} b^{4} x^{3} + 14 \, a^{2} b^{5} x^{2} - 7 \, a b^{6} x + 4 \, b^{7}}{20 \, {\left (a x + b\right )}^{2} b^{8} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^3/x^9,x, algorithm="giac")

[Out]

21*a^5*log(abs(a*x + b))/b^8 - 21*a^5*log(abs(x))/b^8 - 1/20*(420*a^6*b*x^6 + 630*a^5*b^2*x^5 + 140*a^4*b^3*x^
4 - 35*a^3*b^4*x^3 + 14*a^2*b^5*x^2 - 7*a*b^6*x + 4*b^7)/((a*x + b)^2*b^8*x^5)

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maple [A]  time = 0.01, size = 106, normalized size = 0.95 \[ -\frac {a^{5}}{2 \left (a x +b \right )^{2} b^{6}}-\frac {6 a^{5}}{\left (a x +b \right ) b^{7}}-\frac {21 a^{5} \ln \relax (x )}{b^{8}}+\frac {21 a^{5} \ln \left (a x +b \right )}{b^{8}}-\frac {15 a^{4}}{b^{7} x}+\frac {5 a^{3}}{b^{6} x^{2}}-\frac {2 a^{2}}{b^{5} x^{3}}+\frac {3 a}{4 b^{4} x^{4}}-\frac {1}{5 b^{3} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x)^3/x^9,x)

[Out]

-1/5/b^3/x^5+3/4*a/b^4/x^4-2*a^2/b^5/x^3+5*a^3/b^6/x^2-15*a^4/b^7/x-1/2*a^5/b^6/(a*x+b)^2-6*a^5/b^7/(a*x+b)-21
*a^5*ln(x)/b^8+21*a^5*ln(a*x+b)/b^8

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maxima [A]  time = 1.01, size = 119, normalized size = 1.07 \[ -\frac {420 \, a^{6} x^{6} + 630 \, a^{5} b x^{5} + 140 \, a^{4} b^{2} x^{4} - 35 \, a^{3} b^{3} x^{3} + 14 \, a^{2} b^{4} x^{2} - 7 \, a b^{5} x + 4 \, b^{6}}{20 \, {\left (a^{2} b^{7} x^{7} + 2 \, a b^{8} x^{6} + b^{9} x^{5}\right )}} + \frac {21 \, a^{5} \log \left (a x + b\right )}{b^{8}} - \frac {21 \, a^{5} \log \relax (x)}{b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^3/x^9,x, algorithm="maxima")

[Out]

-1/20*(420*a^6*x^6 + 630*a^5*b*x^5 + 140*a^4*b^2*x^4 - 35*a^3*b^3*x^3 + 14*a^2*b^4*x^2 - 7*a*b^5*x + 4*b^6)/(a
^2*b^7*x^7 + 2*a*b^8*x^6 + b^9*x^5) + 21*a^5*log(a*x + b)/b^8 - 21*a^5*log(x)/b^8

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mupad [B]  time = 1.18, size = 113, normalized size = 1.02 \[ \frac {42\,a^5\,\mathrm {atanh}\left (\frac {2\,a\,x}{b}+1\right )}{b^8}-\frac {\frac {1}{5\,b}+\frac {7\,a^2\,x^2}{10\,b^3}-\frac {7\,a^3\,x^3}{4\,b^4}+\frac {7\,a^4\,x^4}{b^5}+\frac {63\,a^5\,x^5}{2\,b^6}+\frac {21\,a^6\,x^6}{b^7}-\frac {7\,a\,x}{20\,b^2}}{a^2\,x^7+2\,a\,b\,x^6+b^2\,x^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^9*(a + b/x)^3),x)

[Out]

(42*a^5*atanh((2*a*x)/b + 1))/b^8 - (1/(5*b) + (7*a^2*x^2)/(10*b^3) - (7*a^3*x^3)/(4*b^4) + (7*a^4*x^4)/b^5 +
(63*a^5*x^5)/(2*b^6) + (21*a^6*x^6)/b^7 - (7*a*x)/(20*b^2))/(a^2*x^7 + b^2*x^5 + 2*a*b*x^6)

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sympy [A]  time = 0.51, size = 116, normalized size = 1.05 \[ \frac {21 a^{5} \left (- \log {\relax (x )} + \log {\left (x + \frac {b}{a} \right )}\right )}{b^{8}} + \frac {- 420 a^{6} x^{6} - 630 a^{5} b x^{5} - 140 a^{4} b^{2} x^{4} + 35 a^{3} b^{3} x^{3} - 14 a^{2} b^{4} x^{2} + 7 a b^{5} x - 4 b^{6}}{20 a^{2} b^{7} x^{7} + 40 a b^{8} x^{6} + 20 b^{9} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)**3/x**9,x)

[Out]

21*a**5*(-log(x) + log(x + b/a))/b**8 + (-420*a**6*x**6 - 630*a**5*b*x**5 - 140*a**4*b**2*x**4 + 35*a**3*b**3*
x**3 - 14*a**2*b**4*x**2 + 7*a*b**5*x - 4*b**6)/(20*a**2*b**7*x**7 + 40*a*b**8*x**6 + 20*b**9*x**5)

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