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

Optimal. Leaf size=15 \[ \frac{\sqrt{b x^2} \log (x)}{x} \]

[Out]

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

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Rubi [A]  time = 0.0013557, antiderivative size = 15, 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, 29} \[ \frac{\sqrt{b x^2} \log (x)}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0013055, size = 14, normalized size = 0.93 \[ \frac{b x \log (x)}{\sqrt{b x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.002, size = 14, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( x \right ) }{x}\sqrt{b{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 1.28096, size = 30, normalized size = 2. \begin{align*} \frac{\sqrt{b x^{2}} \log \left (x\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(b*x**2)/x**2, x)

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Giac [A]  time = 1.18072, size = 12, normalized size = 0.8 \begin{align*} \sqrt{b} \log \left ({\left | x \right |}\right ) \mathrm{sgn}\left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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