3.1635 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^3} \, dx\)

Optimal. Leaf size=53 \[ -\frac{3 b \log (a x+b)}{a^4}+\frac{3 x}{a^3}-\frac{3 x}{2 a^2 \left (a+\frac{b}{x}\right )}-\frac{x}{2 a \left (a+\frac{b}{x}\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.056438, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{3 b \log (a x+b)}{a^4}+\frac{3 x}{a^3}-\frac{3 x}{2 a^2 \left (a+\frac{b}{x}\right )}-\frac{x}{2 a \left (a+\frac{b}{x}\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^(-3),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{x}{2 a \left (a + \frac{b}{x}\right )^{2}} - \frac{3 x}{2 a^{2} \left (a + \frac{b}{x}\right )} + \frac{3 \int \frac{1}{a}\, dx}{a^{2}} - \frac{3 b \log{\left (a x + b \right )}}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x)**3,x)

[Out]

-x/(2*a*(a + b/x)**2) - 3*x/(2*a**2*(a + b/x)) + 3*Integral(1/a, x)/a**2 - 3*b*l
og(a*x + b)/a**4

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Mathematica [A]  time = 0.0730867, size = 40, normalized size = 0.75 \[ -\frac{\frac{b^2 (6 a x+5 b)}{(a x+b)^2}+6 b \log (a x+b)-2 a x}{2 a^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^(-3),x]

[Out]

-(-2*a*x + (b^2*(5*b + 6*a*x))/(b + a*x)^2 + 6*b*Log[b + a*x])/(2*a^4)

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Maple [A]  time = 0.01, size = 49, normalized size = 0.9 \[{\frac{x}{{a}^{3}}}-3\,{\frac{{b}^{2}}{ \left ( ax+b \right ){a}^{4}}}+{\frac{{b}^{3}}{2\, \left ( ax+b \right ) ^{2}{a}^{4}}}-3\,{\frac{b\ln \left ( ax+b \right ) }{{a}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^3,x)

[Out]

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

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Maxima [A]  time = 1.42613, size = 77, normalized size = 1.45 \[ -\frac{6 \, a b^{2} x + 5 \, b^{3}}{2 \,{\left (a^{6} x^{2} + 2 \, a^{5} b x + a^{4} b^{2}\right )}} + \frac{x}{a^{3}} - \frac{3 \, b \log \left (a x + b\right )}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.218106, size = 112, normalized size = 2.11 \[ \frac{2 \, a^{3} x^{3} + 4 \, a^{2} b x^{2} - 4 \, a b^{2} x - 5 \, b^{3} - 6 \,{\left (a^{2} b x^{2} + 2 \, a b^{2} x + b^{3}\right )} \log \left (a x + b\right )}{2 \,{\left (a^{6} x^{2} + 2 \, a^{5} b x + a^{4} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.60726, size = 56, normalized size = 1.06 \[ - \frac{6 a b^{2} x + 5 b^{3}}{2 a^{6} x^{2} + 4 a^{5} b x + 2 a^{4} b^{2}} + \frac{x}{a^{3}} - \frac{3 b \log{\left (a x + b \right )}}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x)**3,x)

[Out]

-(6*a*b**2*x + 5*b**3)/(2*a**6*x**2 + 4*a**5*b*x + 2*a**4*b**2) + x/a**3 - 3*b*l
og(a*x + b)/a**4

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GIAC/XCAS [A]  time = 0.224001, size = 59, normalized size = 1.11 \[ \frac{x}{a^{3}} - \frac{3 \, b{\rm ln}\left ({\left | a x + b \right |}\right )}{a^{4}} - \frac{6 \, a b^{2} x + 5 \, b^{3}}{2 \,{\left (a x + b\right )}^{2} a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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