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

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

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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 {a+b x}}{b} \end {gather*}

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*}

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

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a + b*x],x]

[Out]

(2*Sqrt[a + b*x])/b

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

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/Sqrt[a + b*x],x]

[Out]

(2*Sqrt[a + b*x])/b

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fricas [A]  time = 0.94, size = 12, normalized size = 0.86 \begin {gather*} \frac {2 \, \sqrt {b x + a}}{b} \end {gather*}

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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giac [A]  time = 0.99, size = 12, normalized size = 0.86 \begin {gather*} \frac {2 \, \sqrt {b x + a}}{b} \end {gather*}

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

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

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*x)^(1/2),x)

[Out]

(2*(a + b*x)^(1/2))/b

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sympy [A]  time = 0.07, size = 10, normalized size = 0.71 \begin {gather*} \frac {2 \sqrt {a + b x}}{b} \end {gather*}

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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