3.1626 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^2 x^4} \, dx\)

Optimal. Leaf size=42 \[ -\frac{2 a \log (x)}{b^3}+\frac{2 a \log (a x+b)}{b^3}-\frac{a}{b^2 (a x+b)}-\frac{1}{b^2 x} \]

[Out]

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

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Rubi [A]  time = 0.0678242, antiderivative size = 42, 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 \[ -\frac{2 a \log (x)}{b^3}+\frac{2 a \log (a x+b)}{b^3}-\frac{a}{b^2 (a x+b)}-\frac{1}{b^2 x} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^2*x^4),x]

[Out]

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

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Rubi in Sympy [A]  time = 9.74181, size = 39, normalized size = 0.93 \[ - \frac{a}{b^{2} \left (a x + b\right )} - \frac{2 a \log{\left (x \right )}}{b^{3}} + \frac{2 a \log{\left (a x + b \right )}}{b^{3}} - \frac{1}{b^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x)**2/x**4,x)

[Out]

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

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Mathematica [A]  time = 0.0705966, size = 35, normalized size = 0.83 \[ -\frac{b \left (\frac{a}{a x+b}+\frac{1}{x}\right )-2 a \log (a x+b)+2 a \log (x)}{b^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x)^2*x^4),x]

[Out]

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

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Maple [A]  time = 0.015, size = 43, normalized size = 1. \[ -{\frac{1}{{b}^{2}x}}-{\frac{a}{{b}^{2} \left ( ax+b \right ) }}-2\,{\frac{a\ln \left ( x \right ) }{{b}^{3}}}+2\,{\frac{a\ln \left ( ax+b \right ) }{{b}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^2/x^4,x)

[Out]

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

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Maxima [A]  time = 1.45739, size = 61, normalized size = 1.45 \[ -\frac{2 \, a x + b}{a b^{2} x^{2} + b^{3} x} + \frac{2 \, a \log \left (a x + b\right )}{b^{3}} - \frac{2 \, a \log \left (x\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^2*x^4),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.221001, size = 85, normalized size = 2.02 \[ -\frac{2 \, a b x + b^{2} - 2 \,{\left (a^{2} x^{2} + a b x\right )} \log \left (a x + b\right ) + 2 \,{\left (a^{2} x^{2} + a b x\right )} \log \left (x\right )}{a b^{3} x^{2} + b^{4} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^2*x^4),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.5943, size = 36, normalized size = 0.86 \[ \frac{2 a \left (- \log{\left (x \right )} + \log{\left (x + \frac{b}{a} \right )}\right )}{b^{3}} - \frac{2 a x + b}{a b^{2} x^{2} + b^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x)**2/x**4,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.229185, size = 61, normalized size = 1.45 \[ \frac{2 \, a{\rm ln}\left ({\left | a x + b \right |}\right )}{b^{3}} - \frac{2 \, a{\rm ln}\left ({\left | x \right |}\right )}{b^{3}} - \frac{2 \, a x + b}{{\left (a x^{2} + b x\right )} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^2*x^4),x, algorithm="giac")

[Out]

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