3.1644 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^3 x^9} \, dx\)

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*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 = 0.170024, 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 \[ -\frac{21 a^5 \log (x)}{b^8}+\frac{21 a^5 \log (a x+b)}{b^8}-\frac{6 a^5}{b^7 (a x+b)}-\frac{a^5}{2 b^6 (a x+b)^2}-\frac{15 a^4}{b^7 x}+\frac{5 a^3}{b^6 x^2}-\frac{2 a^2}{b^5 x^3}+\frac{3 a}{4 b^4 x^4}-\frac{1}{5 b^3 x^5} \]

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

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Rubi in Sympy [A]  time = 25.914, size = 110, normalized size = 0.99 \[ - \frac{a^{5}}{2 b^{6} \left (a x + b\right )^{2}} - \frac{6 a^{5}}{b^{7} \left (a x + b\right )} - \frac{21 a^{5} \log{\left (x \right )}}{b^{8}} + \frac{21 a^{5} \log{\left (a x + b \right )}}{b^{8}} - \frac{15 a^{4}}{b^{7} x} + \frac{5 a^{3}}{b^{6} x^{2}} - \frac{2 a^{2}}{b^{5} x^{3}} + \frac{3 a}{4 b^{4} x^{4}} - \frac{1}{5 b^{3} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x)**3/x**9,x)

[Out]

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

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Mathematica [A]  time = 0.165946, 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]

-((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 + 63
0*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 = 0.017, size = 106, normalized size = 1. \[ -{\frac{1}{5\,{b}^{3}{x}^{5}}}+{\frac{3\,a}{4\,{b}^{4}{x}^{4}}}-2\,{\frac{{a}^{2}}{{b}^{5}{x}^{3}}}+5\,{\frac{{a}^{3}}{{b}^{6}{x}^{2}}}-15\,{\frac{{a}^{4}}{{b}^{7}x}}-{\frac{{a}^{5}}{2\,{b}^{6} \left ( ax+b \right ) ^{2}}}-6\,{\frac{{a}^{5}}{{b}^{7} \left ( ax+b \right ) }}-21\,{\frac{{a}^{5}\ln \left ( x \right ) }{{b}^{8}}}+21\,{\frac{{a}^{5}\ln \left ( ax+b \right ) }{{b}^{8}}} \]

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 = 1.44598, size = 161, normalized size = 1.45 \[ -\frac{420 \, a^{6} x^{6} + 630 \, a^{5} b x^{5} + 140 \, a^{4} b^{2} x^{4} - 35 \, a^{3} b^{3} x^{3} + 14 \, a^{2} b^{4} x^{2} - 7 \, a b^{5} x + 4 \, b^{6}}{20 \,{\left (a^{2} b^{7} x^{7} + 2 \, a b^{8} x^{6} + b^{9} x^{5}\right )}} + \frac{21 \, a^{5} \log \left (a x + b\right )}{b^{8}} - \frac{21 \, a^{5} \log \left (x\right )}{b^{8}} \]

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 = 0.227151, size = 220, normalized size = 1.98 \[ -\frac{420 \, a^{6} b x^{6} + 630 \, a^{5} b^{2} x^{5} + 140 \, a^{4} b^{3} x^{4} - 35 \, a^{3} b^{4} x^{3} + 14 \, a^{2} b^{5} x^{2} - 7 \, a b^{6} x + 4 \, b^{7} - 420 \,{\left (a^{7} x^{7} + 2 \, a^{6} b x^{6} + a^{5} b^{2} x^{5}\right )} \log \left (a x + b\right ) + 420 \,{\left (a^{7} x^{7} + 2 \, a^{6} b x^{6} + a^{5} b^{2} x^{5}\right )} \log \left (x\right )}{20 \,{\left (a^{2} b^{8} x^{7} + 2 \, a b^{9} x^{6} + b^{10} x^{5}\right )}} \]

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 = 2.7389, size = 116, normalized size = 1.05 \[ \frac{21 a^{5} \left (- \log{\left (x \right )} + \log{\left (x + \frac{b}{a} \right )}\right )}{b^{8}} - \frac{420 a^{6} x^{6} + 630 a^{5} b x^{5} + 140 a^{4} b^{2} x^{4} - 35 a^{3} b^{3} x^{3} + 14 a^{2} b^{4} x^{2} - 7 a b^{5} x + 4 b^{6}}{20 a^{2} b^{7} x^{7} + 40 a b^{8} x^{6} + 20 b^{9} x^{5}} \]

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/XCAS [A]  time = 0.223901, size = 146, normalized size = 1.32 \[ \frac{21 \, a^{5}{\rm ln}\left ({\left | a x + b \right |}\right )}{b^{8}} - \frac{21 \, a^{5}{\rm ln}\left ({\left | x \right |}\right )}{b^{8}} - \frac{420 \, a^{6} b x^{6} + 630 \, a^{5} b^{2} x^{5} + 140 \, a^{4} b^{3} x^{4} - 35 \, a^{3} b^{4} x^{3} + 14 \, a^{2} b^{5} x^{2} - 7 \, a b^{6} x + 4 \, b^{7}}{20 \,{\left (a x + b\right )}^{2} b^{8} x^{5}} \]

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*ln(abs(a*x + b))/b^8 - 21*a^5*ln(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)