3.2248 \(\int \frac{(1+\sqrt{x})^2}{\sqrt{x}} \, dx\)

Optimal. Leaf size=13 \[ \frac{2}{3} \left (\sqrt{x}+1\right )^3 \]

[Out]

(2*(1 + Sqrt[x])^3)/3

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Rubi [A]  time = 0.0020405, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {261} \[ \frac{2}{3} \left (\sqrt{x}+1\right )^3 \]

Antiderivative was successfully verified.

[In]

Int[(1 + Sqrt[x])^2/Sqrt[x],x]

[Out]

(2*(1 + Sqrt[x])^3)/3

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \frac{\left (1+\sqrt{x}\right )^2}{\sqrt{x}} \, dx &=\frac{2}{3} \left (1+\sqrt{x}\right )^3\\ \end{align*}

Mathematica [A]  time = 0.0023733, size = 20, normalized size = 1.54 \[ \frac{2 x^{3/2}}{3}+2 x+2 \sqrt{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Sqrt[x])^2/Sqrt[x],x]

[Out]

2*Sqrt[x] + 2*x + (2*x^(3/2))/3

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Maple [A]  time = 0.001, size = 15, normalized size = 1.2 \begin{align*}{\frac{2}{3}{x}^{{\frac{3}{2}}}}+2\,x+2\,\sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*x^(3/2)+2*x+2*x^(1/2)

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Maxima [A]  time = 0.993555, size = 12, normalized size = 0.92 \begin{align*} \frac{2}{3} \,{\left (\sqrt{x} + 1\right )}^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(sqrt(x) + 1)^3

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Fricas [A]  time = 1.1896, size = 36, normalized size = 2.77 \begin{align*} \frac{2}{3} \,{\left (x + 3\right )} \sqrt{x} + 2 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(x + 3)*sqrt(x) + 2*x

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Sympy [A]  time = 0.130712, size = 17, normalized size = 1.31 \begin{align*} \frac{2 x^{\frac{3}{2}}}{3} + 2 \sqrt{x} + 2 x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x**(3/2)/3 + 2*sqrt(x) + 2*x

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Giac [A]  time = 1.10494, size = 19, normalized size = 1.46 \begin{align*} \frac{2}{3} \, x^{\frac{3}{2}} + 2 \, x + 2 \, \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*x^(3/2) + 2*x + 2*sqrt(x)