3.182 \(\int \frac{1}{\sqrt{a+b x}} \, dx\)

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

[Out]

(2*Sqrt[a + b*x])/b

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Rubi [A]  time = 0.0013685, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {32} \[ \frac{2 \sqrt{a+b x}}{b} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a + b*x],x]

[Out]

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

Mathematica [A]  time = 0.0034517, size = 14, normalized size = 1. \[ \frac{2 \sqrt{a+b x}}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a + b*x],x]

[Out]

(2*Sqrt[a + b*x])/b

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Maple [A]  time = 0.002, size = 13, normalized size = 0.9 \begin{align*} 2\,{\frac{\sqrt{bx+a}}{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.939872, size = 16, normalized size = 1.14 \begin{align*} \frac{2 \, \sqrt{b x + a}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(b*x + a)/b

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Fricas [A]  time = 1.61068, size = 26, normalized size = 1.86 \begin{align*} \frac{2 \, \sqrt{b x + a}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(b*x + a)/b

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Sympy [A]  time = 0.054909, size = 10, normalized size = 0.71 \begin{align*} \frac{2 \sqrt{a + b x}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(a + b*x)/b

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Giac [A]  time = 1.06621, size = 16, normalized size = 1.14 \begin{align*} \frac{2 \, \sqrt{b x + a}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(b*x + a)/b