3.1826 \(\int \frac {(a+\frac {b}{x^2})^2}{x^3} \, dx\)

Optimal. Leaf size=16 \[ -\frac {\left (a+\frac {b}{x^2}\right )^3}{6 b} \]

[Out]

-1/6*(a+b/x^2)^3/b

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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} \[ -\frac {\left (a+\frac {b}{x^2}\right )^3}{6 b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)^2/x^3,x]

[Out]

-(a + b/x^2)^3/(6*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^2}\right )^2}{x^3} \, dx &=-\frac {\left (a+\frac {b}{x^2}\right )^3}{6 b}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 30, normalized size = 1.88 \[ -\frac {a^2}{2 x^2}-\frac {a b}{2 x^4}-\frac {b^2}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)^2/x^3,x]

[Out]

-1/6*b^2/x^6 - (a*b)/(2*x^4) - a^2/(2*x^2)

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fricas [A]  time = 0.74, size = 24, normalized size = 1.50 \[ -\frac {3 \, a^{2} x^{4} + 3 \, a b x^{2} + b^{2}}{6 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(3*a^2*x^4 + 3*a*b*x^2 + b^2)/x^6

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giac [A]  time = 0.15, size = 24, normalized size = 1.50 \[ -\frac {3 \, a^{2} x^{4} + 3 \, a b x^{2} + b^{2}}{6 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(3*a^2*x^4 + 3*a*b*x^2 + b^2)/x^6

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maple [A]  time = 0.00, size = 25, normalized size = 1.56 \[ -\frac {a^{2}}{2 x^{2}}-\frac {a b}{2 x^{4}}-\frac {b^{2}}{6 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a*b/x^4-1/2*a^2/x^2-1/6*b^2/x^6

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maxima [A]  time = 0.83, size = 14, normalized size = 0.88 \[ -\frac {{\left (a + \frac {b}{x^{2}}\right )}^{3}}{6 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(a + b/x^2)^3/b

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mupad [B]  time = 0.04, size = 26, normalized size = 1.62 \[ -\frac {\frac {a^2\,x^4}{2}+\frac {a\,b\,x^2}{2}+\frac {b^2}{6}}{x^6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x^2)^2/x^3,x)

[Out]

-(b^2/6 + (a^2*x^4)/2 + (a*b*x^2)/2)/x^6

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sympy [B]  time = 0.19, size = 26, normalized size = 1.62 \[ \frac {- 3 a^{2} x^{4} - 3 a b x^{2} - b^{2}}{6 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-3*a**2*x**4 - 3*a*b*x**2 - b**2)/(6*x**6)

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