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

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

[Out]

x+2*x^(1/2)

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Rubi [A]  time = 0.00, antiderivative size = 9, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 9, normalized size = 1.00 \[ x+2 \sqrt {x} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Sqrt[x])/Sqrt[x],x]

[Out]

2*Sqrt[x] + x

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fricas [A]  time = 0.88, size = 7, normalized size = 0.78 \[ x + 2 \, \sqrt {x} \]

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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giac [A]  time = 0.17, size = 7, normalized size = 0.78 \[ x + 2 \, \sqrt {x} \]

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)

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maple [A]  time = 0.00, size = 8, normalized size = 0.89 \[ x +2 \sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x+2*x^(1/2)

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maxima [C]  time = 0.86, size = 7, normalized size = 0.78 \[ {\left (\sqrt {x} + 1\right )}^{2} \]

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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mupad [B]  time = 0.02, size = 7, normalized size = 0.78 \[ x+2\,\sqrt {x} \]

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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sympy [A]  time = 0.15, size = 7, normalized size = 0.78 \[ 2 \sqrt {x} + x \]

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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