3.1839 \(\int \frac{(a+\frac{b}{x^2})^3}{x^3} \, dx\)

Optimal. Leaf size=16 \[ -\frac{\left (a+\frac{b}{x^2}\right )^4}{8 b} \]

[Out]

-(a + b/x^2)^4/(8*b)

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Rubi [A]  time = 0.0037042, 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} \[ -\frac{\left (a+\frac{b}{x^2}\right )^4}{8 b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)^3/x^3,x]

[Out]

-(a + b/x^2)^4/(8*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 )^3}{x^3} \, dx &=-\frac{\left (a+\frac{b}{x^2}\right )^4}{8 b}\\ \end{align*}

Mathematica [B]  time = 0.0063126, size = 43, normalized size = 2.69 \[ -\frac{3 a^2 b}{4 x^4}-\frac{a^3}{2 x^2}-\frac{a b^2}{2 x^6}-\frac{b^3}{8 x^8} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)^3/x^3,x]

[Out]

-b^3/(8*x^8) - (a*b^2)/(2*x^6) - (3*a^2*b)/(4*x^4) - a^3/(2*x^2)

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Maple [B]  time = 0.004, size = 36, normalized size = 2.3 \begin{align*} -{\frac{3\,{a}^{2}b}{4\,{x}^{4}}}-{\frac{{a}^{3}}{2\,{x}^{2}}}-{\frac{{b}^{3}}{8\,{x}^{8}}}-{\frac{{b}^{2}a}{2\,{x}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/4*a^2*b/x^4-1/2*a^3/x^2-1/8*b^3/x^8-1/2*a*b^2/x^6

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Maxima [A]  time = 0.941985, size = 19, normalized size = 1.19 \begin{align*} -\frac{{\left (a + \frac{b}{x^{2}}\right )}^{4}}{8 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(a + b/x^2)^4/b

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Fricas [B]  time = 1.38565, size = 76, normalized size = 4.75 \begin{align*} -\frac{4 \, a^{3} x^{6} + 6 \, a^{2} b x^{4} + 4 \, a b^{2} x^{2} + b^{3}}{8 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(4*a^3*x^6 + 6*a^2*b*x^4 + 4*a*b^2*x^2 + b^3)/x^8

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Sympy [B]  time = 0.403389, size = 37, normalized size = 2.31 \begin{align*} - \frac{4 a^{3} x^{6} + 6 a^{2} b x^{4} + 4 a b^{2} x^{2} + b^{3}}{8 x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(4*a**3*x**6 + 6*a**2*b*x**4 + 4*a*b**2*x**2 + b**3)/(8*x**8)

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Giac [B]  time = 1.17921, size = 47, normalized size = 2.94 \begin{align*} -\frac{4 \, a^{3} x^{6} + 6 \, a^{2} b x^{4} + 4 \, a b^{2} x^{2} + b^{3}}{8 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(4*a^3*x^6 + 6*a^2*b*x^4 + 4*a*b^2*x^2 + b^3)/x^8