\(\int \frac {1}{(a+\frac {b}{x^2})^{3/2} x^2} \, dx\) [1940]

Optimal result

Integrand size = 15, antiderivative size = 19 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{3/2} x^2} \, dx=-\frac {1}{a \sqrt {a+\frac {b}{x^2}} x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 19, 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 = {270} \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{3/2} x^2} \, dx=-\frac {1}{a x \sqrt {a+\frac {b}{x^2}}} \]

[In]

Int[1/((a + b/x^2)^(3/2)*x^2),x]

[Out]

-(1/(a*Sqrt[a + b/x^2]*x))

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {1}{a \sqrt {a+\frac {b}{x^2}} x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{3/2} x^2} \, dx=-\frac {1}{a \sqrt {a+\frac {b}{x^2}} x} \]

[In]

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

[Out]

-(1/(a*Sqrt[a + b/x^2]*x))

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.53

method	result	size
gosper	\(-\frac {a \,x^{2}+b}{a \,x^{3} \left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {3}{2}}}\)	\(29\)
default	\(-\frac {a \,x^{2}+b}{a \,x^{3} \left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {3}{2}}}\)	\(29\)
trager	\(-\frac {x \sqrt {-\frac {-a \,x^{2}-b}{x^{2}}}}{a \left (a \,x^{2}+b \right )}\)	\(33\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.53 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{3/2} x^2} \, dx=-\frac {x \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a^{2} x^{2} + a b} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.47 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{3/2} x^2} \, dx=- \frac {1}{a \sqrt {b} \sqrt {\frac {a x^{2}}{b} + 1}} \]

[In]

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

[Out]

-1/(a*sqrt(b)*sqrt(a*x**2/b + 1))

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{3/2} x^2} \, dx=-\frac {1}{\sqrt {a + \frac {b}{x^{2}}} a x} \]

[In]

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

[Out]

-1/(sqrt(a + b/x^2)*a*x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.47 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{3/2} x^2} \, dx=\frac {\mathrm {sgn}\left (x\right )}{a \sqrt {b}} - \frac {1}{\sqrt {a x^{2} + b} a \mathrm {sgn}\left (x\right )} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.70 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.26 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{3/2} x^2} \, dx=-\frac {x\,\sqrt {a+\frac {b}{x^2}}}{a\,\left (a\,x^2+b\right )} \]

[In]

int(1/(x^2*(a + b/x^2)^(3/2)),x)

[Out]

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