\(\int \frac {1}{\sqrt {a+\frac {b}{x^2}}} \, dx\) [1924]

Optimal result

Integrand size = 11, antiderivative size = 16 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}}} \, dx=\frac {\sqrt {a+\frac {b}{x^2}} x}{a} \]

[Out]

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

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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {197} \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}}} \, dx=\frac {x \sqrt {a+\frac {b}{x^2}}}{a} \]

[In]

Int[1/Sqrt[a + b/x^2],x]

[Out]

(Sqrt[a + b/x^2]*x)/a

Rule 197

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

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}}} \, dx=\frac {\sqrt {a+\frac {b}{x^2}} x}{a} \]

[In]

Integrate[1/Sqrt[a + b/x^2],x]

[Out]

(Sqrt[a + b/x^2]*x)/a

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.44

method	result	size
trager	\(\frac {x \sqrt {-\frac {-a \,x^{2}-b}{x^{2}}}}{a}\)	\(23\)
gosper	\(\frac {a \,x^{2}+b}{a x \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}\)	\(28\)
default	\(\frac {a \,x^{2}+b}{a x \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}\)	\(28\)
risch	\(\frac {a \,x^{2}+b}{a x \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}\)	\(28\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}}} \, dx=\frac {x \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a} \]

[In]

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

[Out]

x*sqrt((a*x^2 + b)/x^2)/a

Sympy [A] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}}} \, dx=\frac {\sqrt {b} \sqrt {\frac {a x^{2}}{b} + 1}}{a} \]

[In]

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

[Out]

sqrt(b)*sqrt(a*x**2/b + 1)/a

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}}} \, dx=\frac {\sqrt {a + \frac {b}{x^{2}}} x}{a} \]

[In]

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

[Out]

sqrt(a + b/x^2)*x/a

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.75 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}}} \, dx=-\frac {\sqrt {b} \mathrm {sgn}\left (x\right )}{a} + \frac {\sqrt {a x^{2} + b}}{a \mathrm {sgn}\left (x\right )} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 6.13 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.44 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}}} \, dx=\frac {x\,\sqrt {\frac {a\,x^2}{b}+1}}{\sqrt {a+\frac {b}{x^2}}\,\left (\sqrt {\frac {a\,x^2}{b}+1}+1\right )} \]

[In]

int(1/(a + b/x^2)^(1/2),x)

[Out]

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