\(\int \frac {1}{(a+\frac {b}{x^2})^{3/2} x^3} \, dx\) [1933]

Optimal result

Integrand size = 15, antiderivative size = 15 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{3/2} x^3} \, dx=\frac {1}{b \sqrt {a+\frac {b}{x^2}}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, 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} \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{3/2} x^3} \, dx=\frac {1}{b \sqrt {a+\frac {b}{x^2}}} \]

[In]

Int[1/((a + b/x^2)^(3/2)*x^3),x]

[Out]

1/(b*Sqrt[a + b/x^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*} \text {integral}& = \frac {1}{b \sqrt {a+\frac {b}{x^2}}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{3/2} x^3} \, dx=\frac {1}{b \sqrt {a+\frac {b}{x^2}}} \]

[In]

Integrate[1/((a + b/x^2)^(3/2)*x^3),x]

[Out]

1/(b*Sqrt[a + b/x^2])

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93

method	result	size
derivativedivides	\(\frac {1}{b \sqrt {a +\frac {b}{x^{2}}}}\)	\(14\)
gosper	\(\frac {a \,x^{2}+b}{x^{2} b \left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {3}{2}}}\)	\(28\)
default	\(\frac {a \,x^{2}+b}{x^{2} b \left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {3}{2}}}\)	\(28\)
trager	\(\frac {x^{2} \sqrt {-\frac {-a \,x^{2}-b}{x^{2}}}}{b \left (a \,x^{2}+b \right )}\)	\(34\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (13) = 26\).

Time = 0.31 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.93 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{3/2} x^3} \, dx=\frac {x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a b x^{2} + b^{2}} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

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

Time = 0.35 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.73 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{3/2} x^3} \, dx=\begin {cases} \frac {1}{b \sqrt {a + \frac {b}{x^{2}}}} & \text {for}\: b \neq 0 \\- \frac {1}{2 a^{\frac {3}{2}} x^{2}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((1/(b*sqrt(a + b/x**2)), Ne(b, 0)), (-1/(2*a**(3/2)*x**2), True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{3/2} x^3} \, dx=\frac {1}{\sqrt {a + \frac {b}{x^{2}}} b} \]

[In]

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

[Out]

1/(sqrt(a + b/x^2)*b)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.20 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{3/2} x^3} \, dx=\frac {x}{\sqrt {a x^{2} + b} b \mathrm {sgn}\left (x\right )} \]

[In]

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

[Out]

x/(sqrt(a*x^2 + b)*b*sgn(x))

Mupad [B] (verification not implemented)

Time = 6.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.20 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{3/2} x^3} \, dx=\frac {\sqrt {x^2}}{b\,\sqrt {a\,x^2+b}} \]

[In]

int(1/(x^3*(a + b/x^2)^(3/2)),x)

[Out]

(x^2)^(1/2)/(b*(b + a*x^2)^(1/2))