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

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

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Rubi [A]  time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {2 \sqrt {c+d x}}{d} \end {gather*}

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*}

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Mathematica [A]  time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {c+d x}}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[c + d*x],x]

[Out]

(2*Sqrt[c + d*x])/d

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IntegrateAlgebraic [A]  time = 0.01, size = 14, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {c+d x}}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/Sqrt[c + d*x],x]

[Out]

(2*Sqrt[c + d*x])/d

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fricas [A]  time = 1.22, size = 12, normalized size = 0.86 \begin {gather*} \frac {2 \, \sqrt {d x + c}}{d} \end {gather*}

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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giac [A]  time = 0.91, size = 12, normalized size = 0.86 \begin {gather*} \frac {2 \, \sqrt {d x + c}}{d} \end {gather*}

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

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maple [A]  time = 0.00, size = 13, normalized size = 0.93 \begin {gather*} \frac {2 \sqrt {d x +c}}{d} \end {gather*}

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 = 1.30, size = 12, normalized size = 0.86 \begin {gather*} \frac {2 \, \sqrt {d x + c}}{d} \end {gather*}

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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mupad [B]  time = 0.02, size = 12, normalized size = 0.86 \begin {gather*} \frac {2\,\sqrt {c+d\,x}}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c + d*x)^(1/2),x)

[Out]

(2*(c + d*x)^(1/2))/d

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sympy [A]  time = 0.06, size = 10, normalized size = 0.71 \begin {gather*} \frac {2 \sqrt {c + d x}}{d} \end {gather*}

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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