3.2247 \(\int \frac{1+\sqrt{x}}{\sqrt{x}} \, dx\)

Optimal. Leaf size=9 \[ x+2 \sqrt{x} \]

[Out]

2*Sqrt[x] + x

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Rubi [A]  time = 0.0025924, antiderivative size = 9, 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} \[ x+2 \sqrt{x} \]

Antiderivative was successfully verified.

[In]

Int[(1 + Sqrt[x])/Sqrt[x],x]

[Out]

2*Sqrt[x] + 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{1+\sqrt{x}}{\sqrt{x}} \, dx &=\int \left (1+\frac{1}{\sqrt{x}}\right ) \, dx\\ &=2 \sqrt{x}+x\\ \end{align*}

Mathematica [A]  time = 0.0016986, size = 9, normalized size = 1. \[ x+2 \sqrt{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Sqrt[x])/Sqrt[x],x]

[Out]

2*Sqrt[x] + x

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Maple [A]  time = 0., size = 8, normalized size = 0.9 \begin{align*} x+2\,\sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x+2*x^(1/2)

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Maxima [A]  time = 0.951112, size = 9, normalized size = 1. \begin{align*}{\left (\sqrt{x} + 1\right )}^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(sqrt(x) + 1)^2

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Fricas [A]  time = 1.22789, size = 20, normalized size = 2.22 \begin{align*} x + 2 \, \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + 2*sqrt(x)

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Sympy [A]  time = 0.124055, size = 7, normalized size = 0.78 \begin{align*} 2 \sqrt{x} + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x) + x

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Giac [A]  time = 1.09503, size = 9, normalized size = 1. \begin{align*} x + 2 \, \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + 2*sqrt(x)