3.2116 \(\int \frac{a+b \sqrt{x}}{x} \, dx\)

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 = 0.0045222, 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, Rules used = {14} \[ a \log (x)+2 b \sqrt{x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])/x,x]

[Out]

2*b*Sqrt[x] + a*Log[x]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \frac{a+b \sqrt{x}}{x} \, dx &=\int \left (\frac{a}{x}+\frac{b}{\sqrt{x}}\right ) \, dx\\ &=2 b \sqrt{x}+a \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0048355, 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., size = 12, normalized size = 0.9 \begin{align*} a\ln \left ( x \right ) +2\,b\sqrt{x} \end{align*}

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 = 0.967913, size = 15, normalized size = 1.15 \begin{align*} a \log \left (x\right ) + 2 \, b \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*log(x) + 2*b*sqrt(x)

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Fricas [A]  time = 1.48151, size = 43, normalized size = 3.31 \begin{align*} 2 \, a \log \left (\sqrt{x}\right ) + 2 \, b \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*log(sqrt(x)) + 2*b*sqrt(x)

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Sympy [A]  time = 0.151494, size = 12, normalized size = 0.92 \begin{align*} a \log{\left (x \right )} + 2 b \sqrt{x} \end{align*}

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 [A]  time = 1.1126, size = 16, normalized size = 1.23 \begin{align*} a \log \left ({\left | x \right |}\right ) + 2 \, b \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*log(abs(x)) + 2*b*sqrt(x)