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

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

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Rubi [A]  time = 0.02, antiderivative size = 53, normalized size of antiderivative = 1.00, 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 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])

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]

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*}

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Mathematica [A]  time = 0.04, 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]

-1/2*(-2*a*x + (b^2*(5*b + 6*a*x))/(b + a*x)^2 + 6*b*Log[b + a*x])/a^4

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fricas [A]  time = 0.74, size = 83, normalized size = 1.57 \[ \frac {2 \, a^{3} x^{3} + 4 \, a^{2} b x^{2} - 4 \, a b^{2} x - 5 \, b^{3} - 6 \, {\left (a^{2} b x^{2} + 2 \, a b^{2} x + b^{3}\right )} \log \left (a x + b\right )}{2 \, {\left (a^{6} x^{2} + 2 \, a^{5} b x + a^{4} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(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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giac [A]  time = 0.15, size = 44, normalized size = 0.83 \[ \frac {x}{a^{3}} - \frac {3 \, b \log \left ({\left | a x + b \right |}\right )}{a^{4}} - \frac {6 \, a b^{2} x + 5 \, b^{3}}{2 \, {\left (a x + b\right )}^{2} a^{4}} \]

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)

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maple [A]  time = 0.01, size = 49, normalized size = 0.92 \[ \frac {b^{3}}{2 \left (a x +b \right )^{2} a^{4}}+\frac {x}{a^{3}}-\frac {3 b^{2}}{\left (a x +b \right ) a^{4}}-\frac {3 b \ln \left (a x +b \right )}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.04, size = 57, normalized size = 1.08 \[ -\frac {6 \, a b^{2} x + 5 \, b^{3}}{2 \, {\left (a^{6} x^{2} + 2 \, a^{5} b x + a^{4} b^{2}\right )}} + \frac {x}{a^{3}} - \frac {3 \, b \log \left (a x + b\right )}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(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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mupad [B]  time = 0.06, size = 59, normalized size = 1.11 \[ \frac {x}{a^3}-\frac {3\,b^2\,x+\frac {5\,b^3}{2\,a}}{a^5\,x^2+2\,a^4\,b\,x+a^3\,b^2}-\frac {3\,b\,\ln \left (b+a\,x\right )}{a^4} \]

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

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sympy [A]  time = 0.27, size = 58, normalized size = 1.09 \[ \frac {- 6 a b^{2} x - 5 b^{3}}{2 a^{6} x^{2} + 4 a^{5} b x + 2 a^{4} b^{2}} + \frac {x}{a^{3}} - \frac {3 b \log {\left (a x + b \right )}}{a^{4}} \]

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*log(a*x + b)/a**4

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