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

Optimal. Leaf size=14 \[ \frac{\log \left (\sqrt{a}+x\right )}{\sqrt{a}} \]

[Out]

Log[Sqrt[a] + x]/Sqrt[a]

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Rubi [A]  time = 0.0021261, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {31} \[ \frac{\log \left (\sqrt{a}+x\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

[In]

Int[(a + Sqrt[a]*x)^(-1),x]

[Out]

Log[Sqrt[a] + x]/Sqrt[a]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{1}{a+\sqrt{a} x} \, dx &=\frac{\log \left (\sqrt{a}+x\right )}{\sqrt{a}}\\ \end{align*}

Mathematica [A]  time = 0.0039626, size = 16, normalized size = 1.14 \[ \frac{\log \left (\sqrt{a} x+a\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + Sqrt[a]*x)^(-1),x]

[Out]

Log[a + Sqrt[a]*x]/Sqrt[a]

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Maple [A]  time = 0.001, size = 13, normalized size = 0.9 \begin{align*}{\ln \left ( a+x\sqrt{a} \right ){\frac{1}{\sqrt{a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.08035, size = 16, normalized size = 1.14 \begin{align*} \frac{\log \left (\sqrt{a} x + a\right )}{\sqrt{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(sqrt(a)*x + a)/sqrt(a)

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Fricas [A]  time = 1.60401, size = 35, normalized size = 2.5 \begin{align*} \frac{\log \left (x + \sqrt{a}\right )}{\sqrt{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x + sqrt(a))/sqrt(a)

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Sympy [A]  time = 0.068242, size = 14, normalized size = 1. \begin{align*} \frac{\log{\left (\sqrt{a} x + a \right )}}{\sqrt{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(sqrt(a)*x + a)/sqrt(a)

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Giac [A]  time = 1.20454, size = 18, normalized size = 1.29 \begin{align*} \frac{\log \left ({\left | \sqrt{a} x + a \right |}\right )}{\sqrt{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs(sqrt(a)*x + a))/sqrt(a)