3.1637 \(\int \frac{1}{(a+\frac{b}{x})^3} \, dx\)

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

[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 = 0.0202364, 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, Rules used = {192, 193, 43} \[ -\frac{3 x}{2 a^2 \left (a+\frac{b}{x}\right )}-\frac{3 b \log (a x+b)}{a^4}+\frac{3 x}{a^3}-\frac{x}{2 a \left (a+\frac{b}{x}\right )^2} \]

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

Rule 192

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p + 1
], 0] && NeQ[p, -1]

Rule 193

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b}, x] && LtQ[n, 0]
&& IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{\left (a+\frac{b}{x}\right )^3} \, dx &=-\frac{x}{2 a \left (a+\frac{b}{x}\right )^2}+\frac{3 \int \frac{1}{\left (a+\frac{b}{x}\right )^2} \, dx}{2 a}\\ &=-\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+\frac{b}{x}} \, dx}{a^2}\\ &=-\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{x}{b+a x} \, dx}{a^2}\\ &=-\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 \left (\frac{1}{a}-\frac{b}{a (b+a x)}\right ) \, dx}{a^2}\\ &=\frac{3 x}{a^3}-\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 b \log (b+a x)}{a^4}\\ \end{align*}

Mathematica [A]  time = 0.037686, 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 = 0.006, size = 49, normalized size = 0.9 \begin{align*}{\frac{x}{{a}^{3}}}-3\,{\frac{b\ln \left ( ax+b \right ) }{{a}^{4}}}+{\frac{{b}^{3}}{2\,{a}^{4} \left ( ax+b \right ) ^{2}}}-3\,{\frac{{b}^{2}}{{a}^{4} \left ( ax+b \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x)^3,x)

[Out]

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

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Maxima [A]  time = 1.10017, size = 77, normalized size = 1.45 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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 = 1.42842, size = 176, normalized size = 3.32 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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 = 0.398499, size = 56, normalized size = 1.06 \begin{align*} - \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}} \end{align*}

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*log(a*x + b)/a**4

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Giac [A]  time = 1.11505, size = 59, normalized size = 1.11 \begin{align*} \frac{x}{a^{3}} - \frac{3 \, b \log \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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