3.1627 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^2 x^5} \, dx\)

Optimal. Leaf size=58 \[ \frac{3 a^2 \log (x)}{b^4}-\frac{3 a^2 \log (a x+b)}{b^4}+\frac{a^2}{b^3 (a x+b)}+\frac{2 a}{b^3 x}-\frac{1}{2 b^2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.0863852, antiderivative size = 58, 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 \[ \frac{3 a^2 \log (x)}{b^4}-\frac{3 a^2 \log (a x+b)}{b^4}+\frac{a^2}{b^3 (a x+b)}+\frac{2 a}{b^3 x}-\frac{1}{2 b^2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^2*x^5),x]

[Out]

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

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Rubi in Sympy [A]  time = 13.0407, size = 56, normalized size = 0.97 \[ \frac{a^{2}}{b^{3} \left (a x + b\right )} + \frac{3 a^{2} \log{\left (x \right )}}{b^{4}} - \frac{3 a^{2} \log{\left (a x + b \right )}}{b^{4}} + \frac{2 a}{b^{3} x} - \frac{1}{2 b^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x)**2/x**5,x)

[Out]

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

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Mathematica [A]  time = 0.0956589, size = 53, normalized size = 0.91 \[ \frac{b \left (\frac{2 a^2}{a x+b}+\frac{4 a}{x}-\frac{b}{x^2}\right )-6 a^2 \log (a x+b)+6 a^2 \log (x)}{2 b^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x)^2*x^5),x]

[Out]

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

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Maple [A]  time = 0.016, size = 57, normalized size = 1. \[ -{\frac{1}{2\,{b}^{2}{x}^{2}}}+2\,{\frac{a}{{b}^{3}x}}+{\frac{{a}^{2}}{{b}^{3} \left ( ax+b \right ) }}+3\,{\frac{{a}^{2}\ln \left ( x \right ) }{{b}^{4}}}-3\,{\frac{{a}^{2}\ln \left ( ax+b \right ) }{{b}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^2/x^5,x)

[Out]

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

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Maxima [A]  time = 1.45135, size = 86, normalized size = 1.48 \[ \frac{6 \, a^{2} x^{2} + 3 \, a b x - b^{2}}{2 \,{\left (a b^{3} x^{3} + b^{4} x^{2}\right )}} - \frac{3 \, a^{2} \log \left (a x + b\right )}{b^{4}} + \frac{3 \, a^{2} \log \left (x\right )}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^2*x^5),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.220625, size = 116, normalized size = 2. \[ \frac{6 \, a^{2} b x^{2} + 3 \, a b^{2} x - b^{3} - 6 \,{\left (a^{3} x^{3} + a^{2} b x^{2}\right )} \log \left (a x + b\right ) + 6 \,{\left (a^{3} x^{3} + a^{2} b x^{2}\right )} \log \left (x\right )}{2 \,{\left (a b^{4} x^{3} + b^{5} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^2*x^5),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.75347, size = 54, normalized size = 0.93 \[ \frac{3 a^{2} \left (\log{\left (x \right )} - \log{\left (x + \frac{b}{a} \right )}\right )}{b^{4}} + \frac{6 a^{2} x^{2} + 3 a b x - b^{2}}{2 a b^{3} x^{3} + 2 b^{4} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x)**2/x**5,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.224183, size = 86, normalized size = 1.48 \[ -\frac{3 \, a^{2}{\rm ln}\left ({\left | a x + b \right |}\right )}{b^{4}} + \frac{3 \, a^{2}{\rm ln}\left ({\left | x \right |}\right )}{b^{4}} + \frac{6 \, a^{2} b x^{2} + 3 \, a b^{2} x - b^{3}}{2 \,{\left (a x + b\right )} b^{4} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^2*x^5),x, algorithm="giac")

[Out]

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