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

Optimal. Leaf size=9 \[ \sqrt{b x^2} \]

[Out]

Sqrt[b*x^2]

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Rubi [A]  time = 0.0008505, antiderivative size = 9, 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, 8} \[ \sqrt{b x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

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

Mathematica [A]  time = 0.0016626, size = 14, normalized size = 1.56 \[ \frac{b x^2}{\sqrt{b x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.001, size = 8, normalized size = 0.9 \begin{align*} \sqrt{b{x}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 1.23001, size = 18, normalized size = 2. \begin{align*} \sqrt{b x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(b*x^2)

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Sympy [A]  time = 0.190571, size = 10, normalized size = 1.11 \begin{align*} \sqrt{b} \sqrt{x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(b)*sqrt(x**2)

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Giac [A]  time = 1.1058, size = 9, normalized size = 1. \begin{align*} \sqrt{b} x \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,x, algorithm="giac")

[Out]

sqrt(b)*x*sgn(x)