\(\int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^3} \, dx\) [1918]

Optimal result

Integrand size = 15, antiderivative size = 16 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^3} \, dx=-\frac {\sqrt {a+\frac {b}{x^2}}}{b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {267} \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^3} \, dx=-\frac {\sqrt {a+\frac {b}{x^2}}}{b} \]

[In]

Int[1/(Sqrt[a + b/x^2]*x^3),x]

[Out]

-(Sqrt[a + b/x^2]/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*} \text {integral}& = -\frac {\sqrt {a+\frac {b}{x^2}}}{b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^3} \, dx=-\frac {\sqrt {a+\frac {b}{x^2}}}{b} \]

[In]

Integrate[1/(Sqrt[a + b/x^2]*x^3),x]

[Out]

-(Sqrt[a + b/x^2]/b)

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94

method	result	size
derivativedivides	\(-\frac {\sqrt {a +\frac {b}{x^{2}}}}{b}\)	\(15\)
trager	\(-\frac {\sqrt {-\frac {-a \,x^{2}-b}{x^{2}}}}{b}\)	\(23\)
gosper	\(-\frac {a \,x^{2}+b}{x^{2} b \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}\)	\(29\)
default	\(-\frac {a \,x^{2}+b}{x^{2} b \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}\)	\(29\)
risch	\(-\frac {a \,x^{2}+b}{x^{2} b \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}\)	\(29\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^3} \, dx=-\frac {\sqrt {\frac {a x^{2} + b}{x^{2}}}}{b} \]

[In]

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

[Out]

-sqrt((a*x^2 + b)/x^2)/b

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 26 vs. \(2 (12) = 24\).

Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.62 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^3} \, dx=\begin {cases} - \frac {\sqrt {a + \frac {b}{x^{2}}}}{b} & \text {for}\: b \neq 0 \\- \frac {1}{2 \sqrt {a} x^{2}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((-sqrt(a + b/x**2)/b, Ne(b, 0)), (-1/(2*sqrt(a)*x**2), True))

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^3} \, dx=-\frac {\sqrt {a + \frac {b}{x^{2}}}}{b} \]

[In]

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

[Out]

-sqrt(a + b/x^2)/b

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (14) = 28\).

Time = 0.30 (sec) , antiderivative size = 34, normalized size of antiderivative = 2.12 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^3} \, dx=\frac {2 \, \sqrt {a}}{{\left ({\left (\sqrt {a} x - \sqrt {a x^{2} + b}\right )}^{2} - b\right )} \mathrm {sgn}\left (x\right )} \]

[In]

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

[Out]

2*sqrt(a)/(((sqrt(a)*x - sqrt(a*x^2 + b))^2 - b)*sgn(x))

Mupad [B] (verification not implemented)

Time = 6.12 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^3} \, dx=-\frac {\sqrt {a+\frac {b}{x^2}}}{b} \]

[In]

int(1/(x^3*(a + b/x^2)^(1/2)),x)

[Out]

-(a + b/x^2)^(1/2)/b