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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, 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 = 0.01, 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]

-1/3*b^2/x^3 - (a*b)/x^2 - a^2/x

________________________________________________________________________________________

fricas [A]  time = 0.93, size = 22, normalized size = 1.38 \[ -\frac {3 \, a^{2} x^{2} + 3 \, a b x + b^{2}}{3 \, x^{3}} \]

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

________________________________________________________________________________________

giac [A]  time = 0.15, size = 14, normalized size = 0.88 \[ -\frac {{\left (a + \frac {b}{x}\right )}^{3}}{3 \, b} \]

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

________________________________________________________________________________________

maple [A]  time = 0.01, size = 25, normalized size = 1.56 \[ -\frac {a^{2}}{x}-\frac {a b}{x^{2}}-\frac {b^{2}}{3 x^{3}} \]

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

________________________________________________________________________________________

maxima [A]  time = 1.00, size = 14, normalized size = 0.88 \[ -\frac {{\left (a + \frac {b}{x}\right )}^{3}}{3 \, b} \]

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

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 22, normalized size = 1.38 \[ -\frac {a^2\,x^2+a\,b\,x+\frac {b^2}{3}}{x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x)^2/x^2,x)

[Out]

-(b^2/3 + a^2*x^2 + a*b*x)/x^3

________________________________________________________________________________________

sympy [B]  time = 0.17, size = 24, normalized size = 1.50 \[ \frac {- 3 a^{2} x^{2} - 3 a b x - b^{2}}{3 x^{3}} \]

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)

________________________________________________________________________________________