3.1.87 \(\int (\frac {b}{x^2})^{3/2} \, dx\) [87]

Optimal. Leaf size=17 \[ -\frac {b \sqrt {\frac {b}{x^2}}}{2 x} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 17, 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 {b \sqrt {\frac {b}{x^2}}}{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b/x^2)^(3/2),x]

[Out]

-1/2*(b*Sqrt[b/x^2])/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*} \int \left (\frac {b}{x^2}\right )^{3/2} \, dx &=\left (b \sqrt {\frac {b}{x^2}} x\right ) \int \frac {1}{x^3} \, dx\\ &=-\frac {b \sqrt {\frac {b}{x^2}}}{2 x}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 14, normalized size = 0.82 \begin {gather*} -\frac {1}{2} \left (\frac {b}{x^2}\right )^{3/2} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b/x^2)^(3/2),x]

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.02, size = 11, normalized size = 0.65

method	result	size
gosper	\(-\frac {x \left (\frac {b}{x^{2}}\right )^{\frac {3}{2}}}{2}\)	\(11\)
default	\(-\frac {x \left (\frac {b}{x^{2}}\right )^{\frac {3}{2}}}{2}\)	\(11\)
risch	\(-\frac {b \sqrt {\frac {b}{x^{2}}}}{2 x}\)	\(14\)
trager	\(\frac {b \left (x -1\right ) \left (x +1\right ) \sqrt {\frac {b}{x^{2}}}}{2 x}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.28, size = 10, normalized size = 0.59 \begin {gather*} -\frac {1}{2} \, x \left (\frac {b}{x^{2}}\right )^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.37, size = 13, normalized size = 0.76 \begin {gather*} -\frac {b \sqrt {\frac {b}{x^{2}}}}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]
time = 0.18, size = 12, normalized size = 0.71 \begin {gather*} - \frac {x \left (\frac {b}{x^{2}}\right )^{\frac {3}{2}}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 1.88, size = 10, normalized size = 0.59 \begin {gather*} -\frac {b^{\frac {3}{2}} \mathrm {sgn}\left (x\right )}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 0.98, size = 13, normalized size = 0.76 \begin {gather*} -\frac {b^{3/2}\,\sqrt {\frac {1}{x^2}}}{2\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________