3.1.88 \(\int (\frac {b}{x^3})^{3/2} \, dx\) [88]

Optimal. Leaf size=17 \[ -\frac {2 b \sqrt {\frac {b}{x^3}}}{7 x^2} \]

[Out]

-2/7*b*(b/x^3)^(1/2)/x^2

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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 {2 b \sqrt {\frac {b}{x^3}}}{7 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*b*Sqrt[b/x^3])/(7*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 \left (\frac {b}{x^3}\right )^{3/2} \, dx &=\left (b \sqrt {\frac {b}{x^3}} x^{3/2}\right ) \int \frac {1}{x^{9/2}} \, dx\\ &=-\frac {2 b \sqrt {\frac {b}{x^3}}}{7 x^2}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 14, normalized size = 0.82 \begin {gather*} -\frac {2}{7} \left (\frac {b}{x^3}\right )^{3/2} x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b/x^3)^(3/2)*x)/7

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Maple [A]
time = 0.03, size = 11, normalized size = 0.65

method	result	size
gosper	\(-\frac {2 x \left (\frac {b}{x^{3}}\right )^{\frac {3}{2}}}{7}\)	\(11\)
default	\(-\frac {2 x \left (\frac {b}{x^{3}}\right )^{\frac {3}{2}}}{7}\)	\(11\)
trager	\(-\frac {2 b \sqrt {\frac {b}{x^{3}}}}{7 x^{2}}\)	\(14\)
risch	\(-\frac {2 b \sqrt {\frac {b}{x^{3}}}}{7 x^{2}}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/7*x*(b/x^3)^(3/2)

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Maxima [A]
time = 0.28, size = 10, normalized size = 0.59 \begin {gather*} -\frac {2}{7} \, x \left (\frac {b}{x^{3}}\right )^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/7*x*(b/x^3)^(3/2)

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Fricas [A]
time = 0.36, size = 13, normalized size = 0.76 \begin {gather*} -\frac {2 \, b \sqrt {\frac {b}{x^{3}}}}{7 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/7*b*sqrt(b/x^3)/x^2

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Sympy [A]
time = 0.17, size = 14, normalized size = 0.82 \begin {gather*} - \frac {2 x \left (\frac {b}{x^{3}}\right )^{\frac {3}{2}}}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x*(b/x**3)**(3/2)/7

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Giac [A]
time = 1.55, size = 13, normalized size = 0.76 \begin {gather*} -\frac {2 \, b^{2}}{7 \, \sqrt {b x} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/7*b^2/(sqrt(b*x)*x^3)

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Mupad [B]
time = 0.97, size = 13, normalized size = 0.76 \begin {gather*} -\frac {2\,b\,\sqrt {\frac {b}{x^3}}}{7\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*b*(b/x^3)^(1/2))/(7*x^2)

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