3.97 \(\int \frac{1}{(b x)^{3/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0012317, antiderivative size = 12, normalized size of antiderivative = 1., 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} \[ -\frac{2}{b \sqrt{b x}} \]

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*}

Mathematica [A]  time = 0.0011217, size = 10, normalized size = 0.83 \[ -\frac{2 x}{(b x)^{3/2}} \]

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 = 0.001, size = 9, normalized size = 0.8 \begin{align*} -2\,{\frac{x}{ \left ( bx \right ) ^{3/2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.964424, size = 14, normalized size = 1.17 \begin{align*} -\frac{2}{\sqrt{b x} b} \end{align*}

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 = 1.65251, size = 30, normalized size = 2.5 \begin{align*} -\frac{2 \, \sqrt{b x}}{b^{2} x} \end{align*}

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 = 0.059365, size = 10, normalized size = 0.83 \begin{align*} - \frac{2}{b \sqrt{b x}} \end{align*}

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 = 1.15002, size = 14, normalized size = 1.17 \begin{align*} -\frac{2}{\sqrt{b x} b} \end{align*}

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)