3.1627 \(\int \frac {1}{(a+\frac {b}{x})^2 x^3} \, dx\)

Optimal. Leaf size=29 \[ -\frac {\log (a x+b)}{b^2}+\frac {1}{b (a x+b)}+\frac {\log (x)}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {263, 44} \[ -\frac {\log (a x+b)}{b^2}+\frac {1}{b (a x+b)}+\frac {\log (x)}{b^2} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x)^2*x^3),x]

[Out]

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

Rule 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 263

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

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 24, normalized size = 0.83 \[ \frac {\frac {b}{a x+b}-\log (a x+b)+\log (x)}{b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x)^2*x^3),x]

[Out]

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

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fricas [A]  time = 1.03, size = 39, normalized size = 1.34 \[ -\frac {{\left (a x + b\right )} \log \left (a x + b\right ) - {\left (a x + b\right )} \log \relax (x) - b}{a b^{2} x + b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^2/x^3,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.15, size = 31, normalized size = 1.07 \[ -\frac {\log \left ({\left | a x + b \right |}\right )}{b^{2}} + \frac {\log \left ({\left | x \right |}\right )}{b^{2}} + \frac {1}{{\left (a x + b\right )} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^2/x^3,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.01, size = 30, normalized size = 1.03 \[ \frac {1}{\left (a x +b \right ) b}+\frac {\ln \relax (x )}{b^{2}}-\frac {\ln \left (a x +b \right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x)^2/x^3,x)

[Out]

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

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maxima [A]  time = 1.08, size = 28, normalized size = 0.97 \[ \frac {1}{a b x + b^{2}} - \frac {\log \left (a x + b\right )}{b^{2}} + \frac {\log \relax (x)}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^2/x^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.04, size = 26, normalized size = 0.90 \[ \frac {1}{b^2+a\,x\,b}-\frac {2\,\mathrm {atanh}\left (\frac {2\,a\,x}{b}+1\right )}{b^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(a + b/x)^2),x)

[Out]

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

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sympy [A]  time = 0.22, size = 22, normalized size = 0.76 \[ \frac {1}{a b x + b^{2}} + \frac {\log {\relax (x )} - \log {\left (x + \frac {b}{a} \right )}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)**2/x**3,x)

[Out]

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

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