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

Optimal. Leaf size=19 \[ -\frac{1}{3 b x^2 \sqrt{b x^2}} \]

[Out]

-1/(3*b*x^2*Sqrt[b*x^2])

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Rubi [A]  time = 0.0017707, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {15, 30} \[ -\frac{1}{3 b x^2 \sqrt{b x^2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(b*x^2)^(3/2)),x]

[Out]

-1/(3*b*x^2*Sqrt[b*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 \frac{1}{x \left (b x^2\right )^{3/2}} \, dx &=\frac{x \int \frac{1}{x^4} \, dx}{b \sqrt{b x^2}}\\ &=-\frac{1}{3 b x^2 \sqrt{b x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0024179, size = 17, normalized size = 0.89 \[ -\frac{b x^2}{3 \left (b x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(b*x^2)^(3/2)),x]

[Out]

-(b*x^2)/(3*(b*x^2)^(5/2))

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Maple [A]  time = 0.002, size = 10, normalized size = 0.5 \begin{align*} -{\frac{1}{3} \left ( b{x}^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.991847, size = 11, normalized size = 0.58 \begin{align*} -\frac{1}{3 \, b^{\frac{3}{2}} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.43212, size = 38, normalized size = 2. \begin{align*} -\frac{\sqrt{b x^{2}}}{3 \, b^{2} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*sqrt(b*x^2)/(b^2*x^4)

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Sympy [A]  time = 0.770041, size = 15, normalized size = 0.79 \begin{align*} - \frac{1}{3 b^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.19822, size = 20, normalized size = 1.05 \begin{align*} -\frac{1}{3 \, \sqrt{b x^{2}} b x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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