3.1566 \(\int \frac{(a+\frac{b}{x})^2}{x^2} \, dx\)

Optimal. Leaf size=16 \[ -\frac{\left (a+\frac{b}{x}\right )^3}{3 b} \]

[Out]

-(a + b/x)^3/(3*b)

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Rubi [A]  time = 0.0034732, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {261} \[ -\frac{\left (a+\frac{b}{x}\right )^3}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^2/x^2,x]

[Out]

-(a + b/x)^3/(3*b)

Rule 261

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

Rubi steps

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

Mathematica [A]  time = 0.0059557, size = 26, normalized size = 1.62 \[ -\frac{a^2}{x}-\frac{a b}{x^2}-\frac{b^2}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^2/x^2,x]

[Out]

-b^2/(3*x^3) - (a*b)/x^2 - a^2/x

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Maple [A]  time = 0.004, size = 25, normalized size = 1.6 \begin{align*} -{\frac{{b}^{2}}{3\,{x}^{3}}}-{\frac{ab}{{x}^{2}}}-{\frac{{a}^{2}}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*b^2/x^3-1/x^2*a*b-a^2/x

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Maxima [A]  time = 0.957856, size = 19, normalized size = 1.19 \begin{align*} -\frac{{\left (a + \frac{b}{x}\right )}^{3}}{3 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(a + b/x)^3/b

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Fricas [A]  time = 1.3659, size = 51, normalized size = 3.19 \begin{align*} -\frac{3 \, a^{2} x^{2} + 3 \, a b x + b^{2}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(3*a^2*x^2 + 3*a*b*x + b^2)/x^3

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Sympy [B]  time = 0.290183, size = 24, normalized size = 1.5 \begin{align*} - \frac{3 a^{2} x^{2} + 3 a b x + b^{2}}{3 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15208, size = 19, normalized size = 1.19 \begin{align*} -\frac{{\left (a + \frac{b}{x}\right )}^{3}}{3 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(a + b/x)^3/b