3.1628 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^2 x^6} \, dx\)

Optimal. Leaf size=69 \[ -\frac{4 a^3 \log (x)}{b^5}+\frac{4 a^3 \log (a x+b)}{b^5}-\frac{a^3}{b^4 (a x+b)}-\frac{3 a^2}{b^4 x}+\frac{a}{b^3 x^2}-\frac{1}{3 b^2 x^3} \]

[Out]

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

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Rubi [A]  time = 0.101989, antiderivative size = 69, 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 \[ -\frac{4 a^3 \log (x)}{b^5}+\frac{4 a^3 \log (a x+b)}{b^5}-\frac{a^3}{b^4 (a x+b)}-\frac{3 a^2}{b^4 x}+\frac{a}{b^3 x^2}-\frac{1}{3 b^2 x^3} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^2*x^6),x]

[Out]

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

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Rubi in Sympy [A]  time = 14.4772, size = 66, normalized size = 0.96 \[ - \frac{a^{3}}{b^{4} \left (a x + b\right )} - \frac{4 a^{3} \log{\left (x \right )}}{b^{5}} + \frac{4 a^{3} \log{\left (a x + b \right )}}{b^{5}} - \frac{3 a^{2}}{b^{4} x} + \frac{a}{b^{3} x^{2}} - \frac{1}{3 b^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x)**2/x**6,x)

[Out]

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

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Mathematica [A]  time = 0.096982, size = 66, normalized size = 0.96 \[ -\frac{-12 a^3 \log (a x+b)+12 a^3 \log (x)+\frac{b \left (12 a^3 x^3+6 a^2 b x^2-2 a b^2 x+b^3\right )}{x^3 (a x+b)}}{3 b^5} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x)^2*x^6),x]

[Out]

-((b*(b^3 - 2*a*b^2*x + 6*a^2*b*x^2 + 12*a^3*x^3))/(x^3*(b + a*x)) + 12*a^3*Log[
x] - 12*a^3*Log[b + a*x])/(3*b^5)

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Maple [A]  time = 0.017, size = 68, normalized size = 1. \[ -{\frac{1}{3\,{b}^{2}{x}^{3}}}+{\frac{a}{{b}^{3}{x}^{2}}}-3\,{\frac{{a}^{2}}{{b}^{4}x}}-{\frac{{a}^{3}}{{b}^{4} \left ( ax+b \right ) }}-4\,{\frac{{a}^{3}\ln \left ( x \right ) }{{b}^{5}}}+4\,{\frac{{a}^{3}\ln \left ( ax+b \right ) }{{b}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^2/x^6,x)

[Out]

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

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Maxima [A]  time = 1.44179, size = 99, normalized size = 1.43 \[ -\frac{12 \, a^{3} x^{3} + 6 \, a^{2} b x^{2} - 2 \, a b^{2} x + b^{3}}{3 \,{\left (a b^{4} x^{4} + b^{5} x^{3}\right )}} + \frac{4 \, a^{3} \log \left (a x + b\right )}{b^{5}} - \frac{4 \, a^{3} \log \left (x\right )}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^2*x^6),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.227271, size = 128, normalized size = 1.86 \[ -\frac{12 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} - 2 \, a b^{3} x + b^{4} - 12 \,{\left (a^{4} x^{4} + a^{3} b x^{3}\right )} \log \left (a x + b\right ) + 12 \,{\left (a^{4} x^{4} + a^{3} b x^{3}\right )} \log \left (x\right )}{3 \,{\left (a b^{5} x^{4} + b^{6} x^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^2*x^6),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.87241, size = 66, normalized size = 0.96 \[ \frac{4 a^{3} \left (- \log{\left (x \right )} + \log{\left (x + \frac{b}{a} \right )}\right )}{b^{5}} - \frac{12 a^{3} x^{3} + 6 a^{2} b x^{2} - 2 a b^{2} x + b^{3}}{3 a b^{4} x^{4} + 3 b^{5} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x)**2/x**6,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.229466, size = 99, normalized size = 1.43 \[ \frac{4 \, a^{3}{\rm ln}\left ({\left | a x + b \right |}\right )}{b^{5}} - \frac{4 \, a^{3}{\rm ln}\left ({\left | x \right |}\right )}{b^{5}} - \frac{12 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} - 2 \, a b^{3} x + b^{4}}{3 \,{\left (a x + b\right )} b^{5} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^2*x^6),x, algorithm="giac")

[Out]

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