\(\int \sqrt {\frac {b}{x^2}} \, dx\) [81]

Optimal result

Integrand size = 9, antiderivative size = 13 \[ \int \sqrt {\frac {b}{x^2}} \, dx=\sqrt {\frac {b}{x^2}} x \log (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {15, 29} \[ \int \sqrt {\frac {b}{x^2}} \, dx=x \sqrt {\frac {b}{x^2}} \log (x) \]

[In]

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

[Out]

Sqrt[b/x^2]*x*Log[x]

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 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\frac {b}{x^2}} x\right ) \int \frac {1}{x} \, dx \\ & = \sqrt {\frac {b}{x^2}} x \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \sqrt {\frac {b}{x^2}} \, dx=\sqrt {\frac {b}{x^2}} x \log (x) \]

[In]

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

[Out]

Sqrt[b/x^2]*x*Log[x]

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92

method	result	size
default	\(x \ln \left (x \right ) \sqrt {\frac {b}{x^{2}}}\)	\(12\)
risch	\(x \ln \left (x \right ) \sqrt {\frac {b}{x^{2}}}\)	\(12\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \sqrt {\frac {b}{x^2}} \, dx=x \sqrt {\frac {b}{x^{2}}} \log \left (x\right ) \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92 \[ \int \sqrt {\frac {b}{x^2}} \, dx=x \sqrt {\frac {b}{x^{2}}} \log {\left (x \right )} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \sqrt {\frac {b}{x^2}} \, dx=x \sqrt {\frac {b}{x^{2}}} \log \left (x\right ) \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int \sqrt {\frac {b}{x^2}} \, dx=\sqrt {b} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (x\right ) \]

[In]

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

[Out]

sqrt(b)*log(abs(x))*sgn(x)

Mupad [F(-1)]

Timed out. \[ \int \sqrt {\frac {b}{x^2}} \, dx=\int \sqrt {\frac {b}{x^2}} \,d x \]

[In]

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

[Out]

int((b/x^2)^(1/2), x)