3.1639 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^3 x^4} \, dx\)

Optimal. Leaf size=43 \[ -\frac{\log (a x+b)}{b^3}+\frac{1}{b^2 (a x+b)}+\frac{1}{2 b (a x+b)^2}+\frac{\log (x)}{b^3} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0645575, antiderivative size = 43, 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 \[ -\frac{\log (a x+b)}{b^3}+\frac{1}{b^2 (a x+b)}+\frac{1}{2 b (a x+b)^2}+\frac{\log (x)}{b^3} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^3*x^4),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 10.4382, size = 37, normalized size = 0.86 \[ \frac{1}{2 b \left (a x + b\right )^{2}} + \frac{1}{b^{2} \left (a x + b\right )} + \frac{\log{\left (x \right )}}{b^{3}} - \frac{\log{\left (a x + b \right )}}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x)**3/x**4,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0518987, size = 37, normalized size = 0.86 \[ \frac{\frac{b (2 a x+3 b)}{(a x+b)^2}-2 \log (a x+b)+2 \log (x)}{2 b^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x)^3*x^4),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.011, size = 42, normalized size = 1. \[{\frac{1}{2\,b \left ( ax+b \right ) ^{2}}}+{\frac{1}{{b}^{2} \left ( ax+b \right ) }}+{\frac{\ln \left ( x \right ) }{{b}^{3}}}-{\frac{\ln \left ( ax+b \right ) }{{b}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^3/x^4,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.43333, size = 69, normalized size = 1.6 \[ \frac{2 \, a x + 3 \, b}{2 \,{\left (a^{2} b^{2} x^{2} + 2 \, a b^{3} x + b^{4}\right )}} - \frac{\log \left (a x + b\right )}{b^{3}} + \frac{\log \left (x\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^3*x^4),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.233208, size = 108, normalized size = 2.51 \[ \frac{2 \, a b x + 3 \, b^{2} - 2 \,{\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \log \left (a x + b\right ) + 2 \,{\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \log \left (x\right )}{2 \,{\left (a^{2} b^{3} x^{2} + 2 \, a b^{4} x + b^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^3*x^4),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 1.70518, size = 46, normalized size = 1.07 \[ \frac{2 a x + 3 b}{2 a^{2} b^{2} x^{2} + 4 a b^{3} x + 2 b^{4}} + \frac{\log{\left (x \right )} - \log{\left (x + \frac{b}{a} \right )}}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x)**3/x**4,x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.235995, size = 58, normalized size = 1.35 \[ -\frac{{\rm ln}\left ({\left | a x + b \right |}\right )}{b^{3}} + \frac{{\rm ln}\left ({\left | x \right |}\right )}{b^{3}} + \frac{2 \, a b x + 3 \, b^{2}}{2 \,{\left (a x + b\right )}^{2} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^3*x^4),x, algorithm="giac")

[Out]

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