3.19.82 \(\int \frac {1}{(a+\frac {b}{x^2})^3 x^3} \, dx\) [1882]

Optimal. Leaf size=16 \[ \frac {1}{4 b \left (a+\frac {b}{x^2}\right )^2} \]

[Out]

1/4/b/(a+b/x^2)^2

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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 {1}{4 b \left (a+\frac {b}{x^2}\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^2)^3*x^3),x]

[Out]

1/(4*b*(a + b/x^2)^2)

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 {1}{\left (a+\frac {b}{x^2}\right )^3 x^3} \, dx &=\frac {1}{4 b \left (a+\frac {b}{x^2}\right )^2}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 24, normalized size = 1.50 \begin {gather*} -\frac {b+2 a x^2}{4 a^2 \left (b+a x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^2)^3*x^3),x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(30\) vs. \(2(14)=28\).
time = 0.03, size = 31, normalized size = 1.94

method	result	size
derivativedivides	\(\frac {1}{4 b \left (\frac {b}{x^{2}}+a \right )^{2}}\)	\(15\)
gosper	\(-\frac {2 a \,x^{2}+b}{4 \left (a \,x^{2}+b \right )^{2} a^{2}}\)	\(23\)
risch	\(\frac {-\frac {x^{2}}{2 a}-\frac {b}{4 a^{2}}}{\left (a \,x^{2}+b \right )^{2}}\)	\(26\)
default	\(\frac {b}{4 a^{2} \left (a \,x^{2}+b \right )^{2}}-\frac {1}{2 a^{2} \left (a \,x^{2}+b \right )}\)	\(31\)
norman	\(\frac {-\frac {x^{4}}{2 a}-\frac {b \,x^{2}}{4 a^{2}}}{x^{2} \left (a \,x^{2}+b \right )^{2}}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b/x^2+a)^3/x^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.30, size = 14, normalized size = 0.88 \begin {gather*} \frac {1}{4 \, {\left (a + \frac {b}{x^{2}}\right )}^{2} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^3/x^3,x, algorithm="maxima")

[Out]

1/4/((a + b/x^2)^2*b)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (14) = 28\).
time = 0.36, size = 36, normalized size = 2.25 \begin {gather*} -\frac {2 \, a x^{2} + b}{4 \, {\left (a^{4} x^{4} + 2 \, a^{3} b x^{2} + a^{2} b^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^3/x^3,x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (12) = 24\).
time = 0.12, size = 36, normalized size = 2.25 \begin {gather*} \frac {- 2 a x^{2} - b}{4 a^{4} x^{4} + 8 a^{3} b x^{2} + 4 a^{2} b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.82, size = 22, normalized size = 1.38 \begin {gather*} -\frac {2 \, a x^{2} + b}{4 \, {\left (a x^{2} + b\right )}^{2} a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^3/x^3,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 1.08, size = 37, normalized size = 2.31 \begin {gather*} -\frac {\frac {b}{4\,a^2}+\frac {x^2}{2\,a}}{a^2\,x^4+2\,a\,b\,x^2+b^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(a + b/x^2)^3),x)

[Out]

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

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