3.1.97 \(\int \frac {1}{(b x)^{3/2}} \, dx\) [97]

Optimal. Leaf size=12 \[ -\frac {2}{b \sqrt {b x}} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {32} \begin {gather*} -\frac {2}{b \sqrt {b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 10, normalized size = 0.83 \begin {gather*} -\frac {2 x}{(b x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.02, size = 11, normalized size = 0.92

method	result	size
gosper	\(-\frac {2 x}{\left (b x \right )^{\frac {3}{2}}}\)	\(9\)
derivativedivides	\(-\frac {2}{b \sqrt {b x}}\)	\(11\)
default	\(-\frac {2}{b \sqrt {b x}}\)	\(11\)
risch	\(-\frac {2}{b \sqrt {b x}}\)	\(11\)
trager	\(-\frac {2 \sqrt {b x}}{b^{2} x}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 10, normalized size = 0.83 \begin {gather*} -\frac {2}{\sqrt {b x} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.35, size = 13, normalized size = 1.08 \begin {gather*} -\frac {2 \, \sqrt {b x}}{b^{2} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.01, size = 10, normalized size = 0.83 \begin {gather*} - \frac {2}{b \sqrt {b x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 2.16, size = 10, normalized size = 0.83 \begin {gather*} -\frac {2}{\sqrt {b x} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.02, size = 10, normalized size = 0.83 \begin {gather*} -\frac {2}{b\,\sqrt {b\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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