\(\int \frac {1}{(a+\frac {b}{x^2})^{5/2} x^3} \, dx\) [1947]

Optimal result

Integrand size = 15, antiderivative size = 18 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{5/2} x^3} \, dx=\frac {1}{3 b \left (a+\frac {b}{x^2}\right )^{3/2}} \]

[Out]

1/3/b/(a+b/x^2)^(3/2)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 18, 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} \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{5/2} x^3} \, dx=\frac {1}{3 b \left (a+\frac {b}{x^2}\right )^{3/2}} \]

[In]

Int[1/((a + b/x^2)^(5/2)*x^3),x]

[Out]

1/(3*b*(a + b/x^2)^(3/2))

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}{3 b \left (a+\frac {b}{x^2}\right )^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.56 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{5/2} x^3} \, dx=\frac {b+a x^2}{3 b \left (a+\frac {b}{x^2}\right )^{5/2} x^2} \]

[In]

Integrate[1/((a + b/x^2)^(5/2)*x^3),x]

[Out]

(b + a*x^2)/(3*b*(a + b/x^2)^(5/2)*x^2)

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83

method	result	size
derivativedivides	\(\frac {1}{3 b \left (a +\frac {b}{x^{2}}\right )^{\frac {3}{2}}}\)	\(15\)
gosper	\(\frac {a \,x^{2}+b}{3 x^{2} b \left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {5}{2}}}\)	\(29\)
default	\(\frac {a \,x^{2}+b}{3 x^{2} b \left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {5}{2}}}\)	\(29\)
trager	\(\frac {x^{4} \sqrt {-\frac {-a \,x^{2}-b}{x^{2}}}}{3 \left (a \,x^{2}+b \right )^{2} b}\)	\(35\)

[In]

int(1/(a+b/x^2)^(5/2)/x^3,x,method=_RETURNVERBOSE)

[Out]

1/3/b/(a+b/x^2)^(3/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (14) = 28\).

Time = 0.30 (sec) , antiderivative size = 41, normalized size of antiderivative = 2.28 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{5/2} x^3} \, dx=\frac {x^{4} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{3 \, {\left (a^{2} b x^{4} + 2 \, a b^{2} x^{2} + b^{3}\right )}} \]

[In]

integrate(1/(a+b/x^2)^(5/2)/x^3,x, algorithm="fricas")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (14) = 28\).

Time = 0.56 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.67 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{5/2} x^3} \, dx=\begin {cases} \frac {1}{3 a b \sqrt {a + \frac {b}{x^{2}}} + \frac {3 b^{2} \sqrt {a + \frac {b}{x^{2}}}}{x^{2}}} & \text {for}\: b \neq 0 \\- \frac {1}{2 a^{\frac {5}{2}} x^{2}} & \text {otherwise} \end {cases} \]

[In]

integrate(1/(a+b/x**2)**(5/2)/x**3,x)

[Out]

Piecewise((1/(3*a*b*sqrt(a + b/x**2) + 3*b**2*sqrt(a + b/x**2)/x**2), Ne(b, 0)), (-1/(2*a**(5/2)*x**2), True))

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{5/2} x^3} \, dx=\frac {1}{3 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} b} \]

[In]

integrate(1/(a+b/x^2)^(5/2)/x^3,x, algorithm="maxima")

[Out]

1/3/((a + b/x^2)^(3/2)*b)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.17 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{5/2} x^3} \, dx=\frac {x^{3}}{3 \, {\left (a x^{2} + b\right )}^{\frac {3}{2}} b \mathrm {sgn}\left (x\right )} \]

[In]

integrate(1/(a+b/x^2)^(5/2)/x^3,x, algorithm="giac")

[Out]

1/3*x^3/((a*x^2 + b)^(3/2)*b*sgn(x))

Mupad [B] (verification not implemented)

Time = 5.91 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{5/2} x^3} \, dx=\frac {1}{3\,b\,{\left (a+\frac {b}{x^2}\right )}^{3/2}} \]

[In]

int(1/(x^3*(a + b/x^2)^(5/2)),x)

[Out]

1/(3*b*(a + b/x^2)^(3/2))