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

Optimal. Leaf size=13 \[ \frac{x \log (x)}{\sqrt{b x^2}} \]

[Out]

(x*Log[x])/Sqrt[b*x^2]

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Rubi [A]  time = 0.0012351, antiderivative size = 13, normalized size of antiderivative = 1., 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, 29} \[ \frac{x \log (x)}{\sqrt{b x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Log[x])/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 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps

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

Mathematica [A]  time = 0.0004672, size = 13, normalized size = 1. \[ \frac{x \log (x)}{\sqrt{b x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Log[x])/Sqrt[b*x^2]

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Maple [A]  time = 0., size = 12, normalized size = 0.9 \begin{align*}{x\ln \left ( x \right ){\frac{1}{\sqrt{b{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.96582, size = 8, normalized size = 0.62 \begin{align*} \frac{\log \left (x\right )}{\sqrt{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x)/sqrt(b)

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Fricas [A]  time = 1.73714, size = 35, normalized size = 2.69 \begin{align*} \frac{\sqrt{b x^{2}} \log \left (x\right )}{b x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b x^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(b*x**2), x)

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Giac [A]  time = 1.25517, size = 26, normalized size = 2. \begin{align*} \frac{\log \left (\sqrt{{\left | b \right |}}{\left | x \right |}{\left | \mathrm{sgn}\left (x\right ) \right |}\right )}{\sqrt{b} \mathrm{sgn}\left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(sqrt(abs(b))*abs(x)*abs(sgn(x)))/(sqrt(b)*sgn(x))