3.38 \(\int \frac{1}{x^3 \sqrt{b x^2}} \, dx\)

Optimal. Leaf size=16 \[ -\frac{1}{3 x^2 \sqrt{b x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0014343, antiderivative size = 16, 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} \[ -\frac{1}{3 x^2 \sqrt{b x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0021913, size = 16, normalized size = 1. \[ -\frac{1}{3 x^2 \sqrt{b x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 13, normalized size = 0.8 \begin{align*} -{\frac{1}{3\,{x}^{2}}{\frac{1}{\sqrt{b{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02403, size = 11, normalized size = 0.69 \begin{align*} -\frac{1}{3 \, \sqrt{b} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.43501, size = 35, normalized size = 2.19 \begin{align*} -\frac{\sqrt{b x^{2}}}{3 \, b x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.770225, size = 19, normalized size = 1.19 \begin{align*} - \frac{1}{3 \sqrt{b} x^{2} \sqrt{x^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.23565, size = 16, normalized size = 1. \begin{align*} -\frac{1}{3 \, \sqrt{b x^{2}} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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