3.2114 \(\int \frac{a+b \sqrt{x}}{x} \, dx\)

Optimal. Leaf size=13 \[ a \log (x)+2 b \sqrt{x} \]

[Out]

2*b*Sqrt[x] + a*Log[x]

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Rubi [A]  time = 0.0143519, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ a \log (x)+2 b \sqrt{x} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])/x,x]

[Out]

2*b*Sqrt[x] + a*Log[x]

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Rubi in Sympy [A]  time = 2.76122, size = 12, normalized size = 0.92 \[ a \log{\left (x \right )} + 2 b \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*log(x) + 2*b*sqrt(x)

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Mathematica [A]  time = 0.00750456, size = 13, normalized size = 1. \[ a \log (x)+2 b \sqrt{x} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*Sqrt[x])/x,x]

[Out]

2*b*Sqrt[x] + a*Log[x]

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Maple [A]  time = 0.004, size = 12, normalized size = 0.9 \[ a\ln \left ( x \right ) +2\,b\sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/2))/x,x)

[Out]

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

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Maxima [A]  time = 1.44106, size = 15, normalized size = 1.15 \[ a \log \left (x\right ) + 2 \, b \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*log(x) + 2*b*sqrt(x)

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Fricas [A]  time = 0.236997, size = 19, normalized size = 1.46 \[ 2 \, a \log \left (\sqrt{x}\right ) + 2 \, b \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a*log(sqrt(x)) + 2*b*sqrt(x)

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Sympy [A]  time = 0.452686, size = 12, normalized size = 0.92 \[ a \log{\left (x \right )} + 2 b \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**(1/2))/x,x)

[Out]

a*log(x) + 2*b*sqrt(x)

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GIAC/XCAS [A]  time = 0.214973, size = 16, normalized size = 1.23 \[ a{\rm ln}\left ({\left | x \right |}\right ) + 2 \, b \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*ln(abs(x)) + 2*b*sqrt(x)