3.19.39 \(\int \frac {(a+\frac {b}{x^2})^3}{x^3} \, dx\) [1839]

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 = {267} \begin {gather*} -\frac {\left (a+\frac {b}{x^2}\right )^4}{8 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)^3/x^3,x]

[Out]

-1/8*(a + b/x^2)^4/b

Rule 267

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] Leaf count is larger than twice the leaf count of optimal. \(43\) vs. \(2(16)=32\).
time = 0.01, size = 43, normalized size = 2.69 \begin {gather*} -\frac {b^3}{8 x^8}-\frac {a b^2}{2 x^6}-\frac {3 a^2 b}{4 x^4}-\frac {a^3}{2 x^2} \end {gather*}

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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(35\) vs. \(2(14)=28\).
time = 0.02, size = 36, normalized size = 2.25

method	result	size
gosper	\(-\frac {4 a^{3} x^{6}+6 a^{2} b \,x^{4}+4 a \,b^{2} x^{2}+b^{3}}{8 x^{8}}\)	\(36\)
default	\(-\frac {3 a^{2} b}{4 x^{4}}-\frac {a \,b^{2}}{2 x^{6}}-\frac {a^{3}}{2 x^{2}}-\frac {b^{3}}{8 x^{8}}\)	\(36\)
norman	\(\frac {-\frac {1}{2} a^{3} x^{6}-\frac {3}{4} a^{2} b \,x^{4}-\frac {1}{2} a \,b^{2} x^{2}-\frac {1}{8} b^{3}}{x^{8}}\)	\(37\)
risch	\(\frac {-\frac {1}{2} a^{3} x^{6}-\frac {3}{4} a^{2} b \,x^{4}-\frac {1}{2} a \,b^{2} x^{2}-\frac {1}{8} b^{3}}{x^{8}}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 14, normalized size = 0.88 \begin {gather*} -\frac {{\left (a + \frac {b}{x^{2}}\right )}^{4}}{8 \, b} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (14) = 28\).
time = 0.36, size = 35, normalized size = 2.19 \begin {gather*} -\frac {4 \, a^{3} x^{6} + 6 \, a^{2} b x^{4} + 4 \, a b^{2} x^{2} + b^{3}}{8 \, x^{8}} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (12) = 24\).
time = 0.11, size = 37, normalized size = 2.31 \begin {gather*} \frac {- 4 a^{3} x^{6} - 6 a^{2} b x^{4} - 4 a b^{2} x^{2} - b^{3}}{8 x^{8}} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (14) = 28\).
time = 0.53, size = 35, normalized size = 2.19 \begin {gather*} -\frac {4 \, a^{3} x^{6} + 6 \, a^{2} b x^{4} + 4 \, a b^{2} x^{2} + b^{3}}{8 \, x^{8}} \end {gather*}

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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Mupad [B]
time = 0.03, size = 37, normalized size = 2.31 \begin {gather*} -\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} \end {gather*}

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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