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]

-1/8*(a+b/x^2)^4/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} \[ -\frac {\left (a+\frac {b}{x^2}\right )^4}{8 b} \]

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*}

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Mathematica [B]  time = 0.01, size = 43, normalized size = 2.69 \[ -\frac {a^3}{2 x^2}-\frac {3 a^2 b}{4 x^4}-\frac {a b^2}{2 x^6}-\frac {b^3}{8 x^8} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)^3/x^3,x]

[Out]

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

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fricas [B]  time = 0.78, size = 35, normalized size = 2.19 \[ -\frac {4 \, a^{3} x^{6} + 6 \, a^{2} b x^{4} + 4 \, a b^{2} x^{2} + b^{3}}{8 \, x^{8}} \]

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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giac [B]  time = 0.15, size = 35, normalized size = 2.19 \[ -\frac {4 \, a^{3} x^{6} + 6 \, a^{2} b x^{4} + 4 \, a b^{2} x^{2} + b^{3}}{8 \, x^{8}} \]

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

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maple [B]  time = 0.00, size = 36, normalized size = 2.25 \[ -\frac {a^{3}}{2 x^{2}}-\frac {3 a^{2} b}{4 x^{4}}-\frac {a \,b^{2}}{2 x^{6}}-\frac {b^{3}}{8 x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.85, size = 14, normalized size = 0.88 \[ -\frac {{\left (a + \frac {b}{x^{2}}\right )}^{4}}{8 \, b} \]

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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mupad [B]  time = 0.03, size = 37, normalized size = 2.31 \[ -\frac {\frac {a^3\,x^6}{2}+\frac {3\,a^2\,b\,x^4}{4}+\frac {a\,b^2\,x^2}{2}+\frac {b^3}{8}}{x^8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x^2)^3/x^3,x)

[Out]

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

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sympy [B]  time = 0.30, size = 37, normalized size = 2.31 \[ \frac {- 4 a^{3} x^{6} - 6 a^{2} b x^{4} - 4 a b^{2} x^{2} - b^{3}}{8 x^{8}} \]

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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