3.79 \(\int \sqrt{b x} \, dx\)

Optimal. Leaf size=14 \[ \frac{2 (b x)^{3/2}}{3 b} \]

[Out]

(2*(b*x)^(3/2))/(3*b)

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Rubi [A]  time = 0.0012822, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {32} \[ \frac{2 (b x)^{3/2}}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[b*x],x]

[Out]

(2*(b*x)^(3/2))/(3*b)

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int \sqrt{b x} \, dx &=\frac{2 (b x)^{3/2}}{3 b}\\ \end{align*}

Mathematica [A]  time = 0.000872, size = 12, normalized size = 0.86 \[ \frac{2}{3} x \sqrt{b x} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[b*x],x]

[Out]

(2*x*Sqrt[b*x])/3

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Maple [A]  time = 0.002, size = 9, normalized size = 0.6 \begin{align*}{\frac{2\,x}{3}\sqrt{bx}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*x*(b*x)^(1/2)

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Maxima [A]  time = 0.968283, size = 14, normalized size = 1. \begin{align*} \frac{2 \, \left (b x\right )^{\frac{3}{2}}}{3 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(b*x)^(3/2)/b

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Fricas [A]  time = 1.80482, size = 23, normalized size = 1.64 \begin{align*} \frac{2}{3} \, \sqrt{b x} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*sqrt(b*x)*x

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Sympy [A]  time = 0.05841, size = 10, normalized size = 0.71 \begin{align*} \frac{2 \left (b x\right )^{\frac{3}{2}}}{3 b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(b*x)**(3/2)/(3*b)

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Giac [A]  time = 1.16492, size = 11, normalized size = 0.79 \begin{align*} \frac{2}{3} \, \sqrt{b x} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*sqrt(b*x)*x