3.1614 \(\int \frac{1}{(a+\frac{b}{x}) x^3} \, dx\)

Optimal. Leaf size=28 \[ -\frac{a \log (x)}{b^2}+\frac{a \log (a x+b)}{b^2}-\frac{1}{b x} \]

[Out]

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

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Rubi [A]  time = 0.0162563, antiderivative size = 28, 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, Rules used = {263, 44} \[ -\frac{a \log (x)}{b^2}+\frac{a \log (a x+b)}{b^2}-\frac{1}{b x} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x)*x^3),x]

[Out]

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

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]

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])

Rubi steps

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

Mathematica [A]  time = 0.0044804, size = 28, normalized size = 1. \[ -\frac{a \log (x)}{b^2}+\frac{a \log (a x+b)}{b^2}-\frac{1}{b x} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x)*x^3),x]

[Out]

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

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Maple [A]  time = 0.006, size = 29, normalized size = 1. \begin{align*} -{\frac{1}{bx}}-{\frac{a\ln \left ( x \right ) }{{b}^{2}}}+{\frac{a\ln \left ( ax+b \right ) }{{b}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.977844, size = 38, normalized size = 1.36 \begin{align*} \frac{a \log \left (a x + b\right )}{b^{2}} - \frac{a \log \left (x\right )}{b^{2}} - \frac{1}{b x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.4317, size = 61, normalized size = 2.18 \begin{align*} \frac{a x \log \left (a x + b\right ) - a x \log \left (x\right ) - b}{b^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.312985, size = 19, normalized size = 0.68 \begin{align*} \frac{a \left (- \log{\left (x \right )} + \log{\left (x + \frac{b}{a} \right )}\right )}{b^{2}} - \frac{1}{b x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.12027, size = 41, normalized size = 1.46 \begin{align*} \frac{a \log \left ({\left | a x + b \right |}\right )}{b^{2}} - \frac{a \log \left ({\left | x \right |}\right )}{b^{2}} - \frac{1}{b x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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