\(\int \sqrt {\frac {b}{x^3}} \, dx\) [82]

Optimal result

Integrand size = 9, antiderivative size = 12 \[ \int \sqrt {\frac {b}{x^3}} \, dx=-2 \sqrt {\frac {b}{x^3}} x \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 12, 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, 30} \[ \int \sqrt {\frac {b}{x^3}} \, dx=-2 x \sqrt {\frac {b}{x^3}} \]

[In]

Int[Sqrt[b/x^3],x]

[Out]

-2*Sqrt[b/x^3]*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 30

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

Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\frac {b}{x^3}} x^{3/2}\right ) \int \frac {1}{x^{3/2}} \, dx \\ & = -2 \sqrt {\frac {b}{x^3}} x \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \sqrt {\frac {b}{x^3}} \, dx=-2 \sqrt {\frac {b}{x^3}} x \]

[In]

Integrate[Sqrt[b/x^3],x]

[Out]

-2*Sqrt[b/x^3]*x

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.92

method	result	size
gosper	\(-2 x \sqrt {\frac {b}{x^{3}}}\)	\(11\)
default	\(-2 x \sqrt {\frac {b}{x^{3}}}\)	\(11\)
trager	\(-2 x \sqrt {\frac {b}{x^{3}}}\)	\(11\)
risch	\(-2 x \sqrt {\frac {b}{x^{3}}}\)	\(11\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \sqrt {\frac {b}{x^3}} \, dx=-2 \, x \sqrt {\frac {b}{x^{3}}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \sqrt {\frac {b}{x^3}} \, dx=- 2 x \sqrt {\frac {b}{x^{3}}} \]

[In]

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

[Out]

-2*x*sqrt(b/x**3)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \sqrt {\frac {b}{x^3}} \, dx=-2 \, x \sqrt {\frac {b}{x^{3}}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int \sqrt {\frac {b}{x^3}} \, dx=-\frac {2 \, b}{\sqrt {b x}} \]

[In]

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

[Out]

-2*b/sqrt(b*x)

Mupad [B] (verification not implemented)

Time = 5.46 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \sqrt {\frac {b}{x^3}} \, dx=-2\,x\,\sqrt {\frac {b}{x^3}} \]

[In]

int((b/x^3)^(1/2),x)

[Out]

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