3.1636 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^3 x} \, dx\)

Optimal. Leaf size=41 \[ -\frac{b^2}{2 a^3 (a x+b)^2}+\frac{2 b}{a^3 (a x+b)}+\frac{\log (a x+b)}{a^3} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0636248, antiderivative size = 41, 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 \[ -\frac{b^2}{2 a^3 (a x+b)^2}+\frac{2 b}{a^3 (a x+b)}+\frac{\log (a x+b)}{a^3} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^3*x),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 11.0221, size = 36, normalized size = 0.88 \[ - \frac{b^{2}}{2 a^{3} \left (a x + b\right )^{2}} + \frac{2 b}{a^{3} \left (a x + b\right )} + \frac{\log{\left (a x + b \right )}}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0243065, size = 33, normalized size = 0.8 \[ \frac{\frac{b (4 a x+3 b)}{(a x+b)^2}+2 \log (a x+b)}{2 a^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x)^3*x),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 40, normalized size = 1. \[ -{\frac{{b}^{2}}{2\,{a}^{3} \left ( ax+b \right ) ^{2}}}+2\,{\frac{b}{{a}^{3} \left ( ax+b \right ) }}+{\frac{\ln \left ( ax+b \right ) }{{a}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^3/x,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.42349, size = 65, normalized size = 1.59 \[ \frac{4 \, a b x + 3 \, b^{2}}{2 \,{\left (a^{5} x^{2} + 2 \, a^{4} b x + a^{3} b^{2}\right )}} + \frac{\log \left (a x + b\right )}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.223915, size = 82, normalized size = 2. \[ \frac{4 \, a b x + 3 \, b^{2} + 2 \,{\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \log \left (a x + b\right )}{2 \,{\left (a^{5} x^{2} + 2 \, a^{4} b x + a^{3} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 1.39151, size = 46, normalized size = 1.12 \[ \frac{4 a b x + 3 b^{2}}{2 a^{5} x^{2} + 4 a^{4} b x + 2 a^{3} b^{2}} + \frac{\log{\left (a x + b \right )}}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x)**3/x,x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.224712, size = 50, normalized size = 1.22 \[ \frac{{\rm ln}\left ({\left | a x + b \right |}\right )}{a^{3}} + \frac{4 \, b x + \frac{3 \, b^{2}}{a}}{2 \,{\left (a x + b\right )}^{2} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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