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

Optimal. Leaf size=14 \[ \frac{1}{2} x \sqrt{b x^2} \]

[Out]

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

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Rubi [A]  time = 0.0015243, antiderivative size = 14, 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, 30} \[ \frac{1}{2} x \sqrt{b x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0006993, size = 14, normalized size = 1. \[ \frac{1}{2} x \sqrt{b x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.001, size = 11, normalized size = 0.8 \begin{align*}{\frac{x}{2}\sqrt{b{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.24049, size = 26, normalized size = 1.86 \begin{align*} \frac{1}{2} \, \sqrt{b x^{2}} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.144732, size = 14, normalized size = 1. \begin{align*} \frac{\sqrt{b} x \sqrt{x^{2}}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.17177, size = 14, normalized size = 1. \begin{align*} \frac{1}{2} \, \sqrt{b} x^{2} \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, algorithm="giac")

[Out]

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