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

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

[Out]

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

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Rubi [A]  time = 0.0012854, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {15, 8} \[ \frac{x^2}{\sqrt{b x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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 8

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

Rubi steps

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

Mathematica [A]  time = 0.0006127, size = 13, normalized size = 1. \[ \frac{x^2}{\sqrt{b x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.001, size = 12, normalized size = 0.9 \begin{align*}{{x}^{2}{\frac{1}{\sqrt{b{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02112, size = 15, normalized size = 1.15 \begin{align*} \frac{\sqrt{b x^{2}}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(b*x^2)/b

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Fricas [A]  time = 1.26161, size = 20, normalized size = 1.54 \begin{align*} \frac{\sqrt{b x^{2}}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(b*x^2)/b

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Sympy [A]  time = 0.42279, size = 14, normalized size = 1.08 \begin{align*} \frac{x^{2}}{\sqrt{b} \sqrt{x^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.16045, size = 15, normalized size = 1.15 \begin{align*} \frac{\sqrt{b x^{2}}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(b*x^2)/b