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

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

[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

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Rubi [A]  time = 0.0671547, antiderivative size = 111, 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, 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 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]

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])

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*}

Mathematica [A]  time = 0.0844765, size = 101, normalized size = 0.91 \[ -\frac{\frac{b \left (14 a^2 b^4 x^2-35 a^3 b^3 x^3+140 a^4 b^2 x^4+630 a^5 b x^5+420 a^6 x^6-7 a b^5 x+4 b^6\right )}{x^5 (a x+b)^2}-420 a^5 \log (a x+b)+420 a^5 \log (x)}{20 b^8} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x)^3*x^9),x]

[Out]

-((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])/(20*b^8)

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Maple [A]  time = 0.012, size = 106, normalized size = 1. \begin{align*} -{\frac{1}{5\,{b}^{3}{x}^{5}}}+{\frac{3\,a}{4\,{b}^{4}{x}^{4}}}-2\,{\frac{{a}^{2}}{{b}^{5}{x}^{3}}}+5\,{\frac{{a}^{3}}{{b}^{6}{x}^{2}}}-15\,{\frac{{a}^{4}}{{b}^{7}x}}-{\frac{{a}^{5}}{2\,{b}^{6} \left ( ax+b \right ) ^{2}}}-6\,{\frac{{a}^{5}}{{b}^{7} \left ( ax+b \right ) }}-21\,{\frac{{a}^{5}\ln \left ( x \right ) }{{b}^{8}}}+21\,{\frac{{a}^{5}\ln \left ( ax+b \right ) }{{b}^{8}}} \end{align*}

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 = 0.988172, size = 161, normalized size = 1.45 \begin{align*} -\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 \left (x\right )}{b^{8}} \end{align*}

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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Fricas [A]  time = 1.69309, size = 351, normalized size = 3.16 \begin{align*} -\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 \left (x\right )}{20 \,{\left (a^{2} b^{8} x^{7} + 2 \, a b^{9} x^{6} + b^{10} x^{5}\right )}} \end{align*}

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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Sympy [A]  time = 0.686328, size = 116, normalized size = 1.05 \begin{align*} \frac{21 a^{5} \left (- \log{\left (x \right )} + \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}} \end{align*}

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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Giac [A]  time = 1.12079, size = 146, normalized size = 1.32 \begin{align*} \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}} \end{align*}

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)