3.1418 \(\int \frac{1}{\sqrt{c+d x}} \, dx\)

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

[Out]

(2*Sqrt[c + d*x])/d

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Rubi [A]  time = 0.0013672, 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} \[ \frac{2 \sqrt{c+d x}}{d} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[c + d*x],x]

[Out]

(2*Sqrt[c + d*x])/d

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{c+d x}} \, dx &=\frac{2 \sqrt{c+d x}}{d}\\ \end{align*}

Mathematica [A]  time = 0.0033623, size = 14, normalized size = 1. \[ \frac{2 \sqrt{c+d x}}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[c + d*x],x]

[Out]

(2*Sqrt[c + d*x])/d

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Maple [A]  time = 0.002, size = 13, normalized size = 0.9 \begin{align*} 2\,{\frac{\sqrt{dx+c}}{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*x+c)^(1/2),x)

[Out]

2*(d*x+c)^(1/2)/d

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Maxima [A]  time = 0.946391, size = 16, normalized size = 1.14 \begin{align*} \frac{2 \, \sqrt{d x + c}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

2*sqrt(d*x + c)/d

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Fricas [A]  time = 1.77387, size = 26, normalized size = 1.86 \begin{align*} \frac{2 \, \sqrt{d x + c}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

2*sqrt(d*x + c)/d

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Sympy [A]  time = 0.05371, size = 10, normalized size = 0.71 \begin{align*} \frac{2 \sqrt{c + d x}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)**(1/2),x)

[Out]

2*sqrt(c + d*x)/d

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Giac [A]  time = 1.0456, size = 16, normalized size = 1.14 \begin{align*} \frac{2 \, \sqrt{d x + c}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^(1/2),x, algorithm="giac")

[Out]

2*sqrt(d*x + c)/d