3.1641 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^3 x^6} \, dx\)

Optimal. Leaf size=76 \[ \frac{6 a^2 \log (x)}{b^5}-\frac{6 a^2 \log (a x+b)}{b^5}+\frac{3 a^2}{b^4 (a x+b)}+\frac{a^2}{2 b^3 (a x+b)^2}+\frac{3 a}{b^4 x}-\frac{1}{2 b^3 x^2} \]

[Out]

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

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Rubi [A]  time = 0.109722, antiderivative size = 76, 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 \[ \frac{6 a^2 \log (x)}{b^5}-\frac{6 a^2 \log (a x+b)}{b^5}+\frac{3 a^2}{b^4 (a x+b)}+\frac{a^2}{2 b^3 (a x+b)^2}+\frac{3 a}{b^4 x}-\frac{1}{2 b^3 x^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^3*x^6),x]

[Out]

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

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Rubi in Sympy [A]  time = 16.4262, size = 73, normalized size = 0.96 \[ \frac{a^{2}}{2 b^{3} \left (a x + b\right )^{2}} + \frac{3 a^{2}}{b^{4} \left (a x + b\right )} + \frac{6 a^{2} \log{\left (x \right )}}{b^{5}} - \frac{6 a^{2} \log{\left (a x + b \right )}}{b^{5}} + \frac{3 a}{b^{4} x} - \frac{1}{2 b^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x)**3/x**6,x)

[Out]

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

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Mathematica [A]  time = 0.104586, size = 68, normalized size = 0.89 \[ \frac{-12 a^2 \log (a x+b)+12 a^2 \log (x)+\frac{b \left (12 a^3 x^3+18 a^2 b x^2+4 a b^2 x-b^3\right )}{x^2 (a x+b)^2}}{2 b^5} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x)^3*x^6),x]

[Out]

((b*(-b^3 + 4*a*b^2*x + 18*a^2*b*x^2 + 12*a^3*x^3))/(x^2*(b + a*x)^2) + 12*a^2*L
og[x] - 12*a^2*Log[b + a*x])/(2*b^5)

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Maple [A]  time = 0.016, size = 73, normalized size = 1. \[ -{\frac{1}{2\,{b}^{3}{x}^{2}}}+3\,{\frac{a}{{b}^{4}x}}+{\frac{{a}^{2}}{2\,{b}^{3} \left ( ax+b \right ) ^{2}}}+3\,{\frac{{a}^{2}}{{b}^{4} \left ( ax+b \right ) }}+6\,{\frac{{a}^{2}\ln \left ( x \right ) }{{b}^{5}}}-6\,{\frac{{a}^{2}\ln \left ( ax+b \right ) }{{b}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^3/x^6,x)

[Out]

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

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Maxima [A]  time = 1.44425, size = 116, normalized size = 1.53 \[ \frac{12 \, a^{3} x^{3} + 18 \, a^{2} b x^{2} + 4 \, a b^{2} x - b^{3}}{2 \,{\left (a^{2} b^{4} x^{4} + 2 \, a b^{5} x^{3} + b^{6} x^{2}\right )}} - \frac{6 \, a^{2} \log \left (a x + b\right )}{b^{5}} + \frac{6 \, a^{2} \log \left (x\right )}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^3*x^6),x, algorithm="maxima")

[Out]

1/2*(12*a^3*x^3 + 18*a^2*b*x^2 + 4*a*b^2*x - b^3)/(a^2*b^4*x^4 + 2*a*b^5*x^3 + b
^6*x^2) - 6*a^2*log(a*x + b)/b^5 + 6*a^2*log(x)/b^5

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Fricas [A]  time = 0.229501, size = 176, normalized size = 2.32 \[ \frac{12 \, a^{3} b x^{3} + 18 \, a^{2} b^{2} x^{2} + 4 \, a b^{3} x - b^{4} - 12 \,{\left (a^{4} x^{4} + 2 \, a^{3} b x^{3} + a^{2} b^{2} x^{2}\right )} \log \left (a x + b\right ) + 12 \,{\left (a^{4} x^{4} + 2 \, a^{3} b x^{3} + a^{2} b^{2} x^{2}\right )} \log \left (x\right )}{2 \,{\left (a^{2} b^{5} x^{4} + 2 \, a b^{6} x^{3} + b^{7} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^3*x^6),x, algorithm="fricas")

[Out]

1/2*(12*a^3*b*x^3 + 18*a^2*b^2*x^2 + 4*a*b^3*x - b^4 - 12*(a^4*x^4 + 2*a^3*b*x^3
+ a^2*b^2*x^2)*log(a*x + b) + 12*(a^4*x^4 + 2*a^3*b*x^3 + a^2*b^2*x^2)*log(x))/
(a^2*b^5*x^4 + 2*a*b^6*x^3 + b^7*x^2)

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Sympy [A]  time = 2.06232, size = 78, normalized size = 1.03 \[ \frac{6 a^{2} \left (\log{\left (x \right )} - \log{\left (x + \frac{b}{a} \right )}\right )}{b^{5}} + \frac{12 a^{3} x^{3} + 18 a^{2} b x^{2} + 4 a b^{2} x - b^{3}}{2 a^{2} b^{4} x^{4} + 4 a b^{5} x^{3} + 2 b^{6} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x)**3/x**6,x)

[Out]

6*a**2*(log(x) - log(x + b/a))/b**5 + (12*a**3*x**3 + 18*a**2*b*x**2 + 4*a*b**2*
x - b**3)/(2*a**2*b**4*x**4 + 4*a*b**5*x**3 + 2*b**6*x**2)

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GIAC/XCAS [A]  time = 0.222919, size = 99, normalized size = 1.3 \[ -\frac{6 \, a^{2}{\rm ln}\left ({\left | a x + b \right |}\right )}{b^{5}} + \frac{6 \, a^{2}{\rm ln}\left ({\left | x \right |}\right )}{b^{5}} + \frac{12 \, a^{3} x^{3} + 18 \, a^{2} b x^{2} + 4 \, a b^{2} x - b^{3}}{2 \,{\left (a x^{2} + b x\right )}^{2} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^3*x^6),x, algorithm="giac")

[Out]

-6*a^2*ln(abs(a*x + b))/b^5 + 6*a^2*ln(abs(x))/b^5 + 1/2*(12*a^3*x^3 + 18*a^2*b*
x^2 + 4*a*b^2*x - b^3)/((a*x^2 + b*x)^2*b^4)