3.3.73 \(\int \frac {1}{a+\sqrt {a} x} \, dx\) [273]

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

[Out]

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

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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 = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {31} \begin {gather*} \frac {\log \left (\sqrt {a}+x\right )}{\sqrt {a}} \end {gather*}

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*}

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Mathematica [A]
time = 0.00, size = 16, normalized size = 1.14 \begin {gather*} \frac {\log \left (a+\sqrt {a} x\right )}{\sqrt {a}} \end {gather*}

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 = 0.09, size = 13, normalized size = 0.93

method	result	size
default	\(\frac {\ln \left (a +x \sqrt {a}\right )}{\sqrt {a}}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 12, normalized size = 0.86 \begin {gather*} \frac {\log \left (\sqrt {a} x + a\right )}{\sqrt {a}} \end {gather*}

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 = 0.42, size = 10, normalized size = 0.71 \begin {gather*} \frac {\log \left (x + \sqrt {a}\right )}{\sqrt {a}} \end {gather*}

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 = 0.01, size = 14, normalized size = 1.00 \begin {gather*} \frac {\log {\left (\sqrt {a} x + a \right )}}{\sqrt {a}} \end {gather*}

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 = 0.71, size = 13, normalized size = 0.93 \begin {gather*} \frac {\log \left ({\left | \sqrt {a} x + a \right |}\right )}{\sqrt {a}} \end {gather*}

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)

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Mupad [B]
time = 0.11, size = 10, normalized size = 0.71 \begin {gather*} \frac {\ln \left (x+\sqrt {a}\right )}{\sqrt {a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + a^(1/2)*x),x)

[Out]

log(x + a^(1/2))/a^(1/2)

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