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 = 0.00783638, 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 \[ \frac{\log \left (\sqrt{a}+x\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

[In]  Int[(a + Sqrt[a]*x)^(-1),x]

[Out]

Log[Sqrt[a] + x]/Sqrt[a]

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Rubi in Sympy [A]  time = 1.25852, size = 12, normalized size = 0.86 \[ \frac{\log{\left (\sqrt{a} + x \right )}}{\sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+x*a**(1/2)),x)

[Out]

log(sqrt(a) + x)/sqrt(a)

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Mathematica [A]  time = 0.00512005, 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., size = 13, normalized size = 0.9 \[{1\ln \left ( a+x\sqrt{a} \right ){\frac{1}{\sqrt{a}}}} \]

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 = 1.35896, size = 16, normalized size = 1.14 \[ \frac{\log \left (\sqrt{a} x + a\right )}{\sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a)*x + a),x, algorithm="maxima")

[Out]

log(sqrt(a)*x + a)/sqrt(a)

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Fricas [A]  time = 0.218018, size = 16, normalized size = 1.14 \[ \frac{\log \left (\sqrt{a} x + a\right )}{\sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a)*x + a),x, algorithm="fricas")

[Out]

log(sqrt(a)*x + a)/sqrt(a)

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Sympy [A]  time = 0.084568, size = 14, normalized size = 1. \[ \frac{\log{\left (\sqrt{a} x + a \right )}}{\sqrt{a}} \]

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/XCAS [A]  time = 0.204295, size = 18, normalized size = 1.29 \[ \frac{{\rm ln}\left ({\left | \sqrt{a} x + a \right |}\right )}{\sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a)*x + a),x, algorithm="giac")

[Out]

ln(abs(sqrt(a)*x + a))/sqrt(a)