3.1.93 \(\int \frac {1}{\sqrt {\frac {b}{x^2}}} \, dx\) [93]

Optimal. Leaf size=14 \[ \frac {x}{2 \sqrt {\frac {b}{x^2}}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {15, 30} \begin {gather*} \frac {x}{2 \sqrt {\frac {b}{x^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[b/x^2],x]

[Out]

x/(2*Sqrt[b/x^2])

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[a^IntPart[m]*((a*x^n)^FracPart[m]/x^(n*FracPart[m])), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {\frac {b}{x^2}}} \, dx &=\frac {\int x \, dx}{\sqrt {\frac {b}{x^2}} x}\\ &=\frac {x}{2 \sqrt {\frac {b}{x^2}}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} \frac {x}{2 \sqrt {\frac {b}{x^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[b/x^2],x]

[Out]

x/(2*Sqrt[b/x^2])

________________________________________________________________________________________

Maple [A]
time = 0.02, size = 11, normalized size = 0.79

method	result	size
gosper	\(\frac {x}{2 \sqrt {\frac {b}{x^{2}}}}\)	\(11\)
default	\(\frac {x}{2 \sqrt {\frac {b}{x^{2}}}}\)	\(11\)
risch	\(\frac {x}{2 \sqrt {\frac {b}{x^{2}}}}\)	\(11\)
trager	\(\frac {\left (x +1\right ) x \left (x -1\right ) \sqrt {\frac {b}{x^{2}}}}{2 b}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.26, size = 10, normalized size = 0.71 \begin {gather*} \frac {x}{2 \, \sqrt {\frac {b}{x^{2}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x/sqrt(b/x^2)

________________________________________________________________________________________

Fricas [A]
time = 0.34, size = 15, normalized size = 1.07 \begin {gather*} \frac {x^{3} \sqrt {\frac {b}{x^{2}}}}{2 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^3*sqrt(b/x^2)/b

________________________________________________________________________________________

Sympy [A]
time = 0.16, size = 10, normalized size = 0.71 \begin {gather*} \frac {x}{2 \sqrt {\frac {b}{x^{2}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/(2*sqrt(b/x**2))

________________________________________________________________________________________

Giac [A]
time = 1.98, size = 12, normalized size = 0.86 \begin {gather*} \frac {x^{2}}{2 \, \sqrt {b} \mathrm {sgn}\left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2/(sqrt(b)*sgn(x))

________________________________________________________________________________________

Mupad [B]
time = 0.93, size = 13, normalized size = 0.93 \begin {gather*} \frac {x^3\,\sqrt {\frac {1}{x^2}}}{2\,\sqrt {b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b/x^2)^(1/2),x)

[Out]

(x^3*(1/x^2)^(1/2))/(2*b^(1/2))

________________________________________________________________________________________