3.2249 \(\int \frac{(1+\sqrt{x})^3}{\sqrt{x}} \, dx\)

Optimal. Leaf size=13 \[ \frac{1}{2} \left (\sqrt{x}+1\right )^4 \]

[Out]

(1 + Sqrt[x])^4/2

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Rubi [A]  time = 0.0022539, 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} \[ \frac{1}{2} \left (\sqrt{x}+1\right )^4 \]

Antiderivative was successfully verified.

[In]

Int[(1 + Sqrt[x])^3/Sqrt[x],x]

[Out]

(1 + Sqrt[x])^4/2

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 )^3}{\sqrt{x}} \, dx &=\frac{1}{2} \left (1+\sqrt{x}\right )^4\\ \end{align*}

Mathematica [A]  time = 0.0024674, size = 13, normalized size = 1. \[ \frac{1}{2} \left (\sqrt{x}+1\right )^4 \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Sqrt[x])^3/Sqrt[x],x]

[Out]

(1 + Sqrt[x])^4/2

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Maple [B]  time = 0.001, size = 20, normalized size = 1.5 \begin{align*}{\frac{{x}^{2}}{2}}+2\,{x}^{3/2}+3\,x+2\,\sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^(1/2)+1)^3/x^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.989979, size = 12, normalized size = 0.92 \begin{align*} \frac{1}{2} \,{\left (\sqrt{x} + 1\right )}^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x^(1/2))^3/x^(1/2),x, algorithm="maxima")

[Out]

1/2*(sqrt(x) + 1)^4

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Fricas [A]  time = 1.23182, size = 47, normalized size = 3.62 \begin{align*} \frac{1}{2} \, x^{2} + 2 \,{\left (x + 1\right )} \sqrt{x} + 3 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x^(1/2))^3/x^(1/2),x, algorithm="fricas")

[Out]

1/2*x^2 + 2*(x + 1)*sqrt(x) + 3*x

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Sympy [B]  time = 0.163897, size = 20, normalized size = 1.54 \begin{align*} 2 x^{\frac{3}{2}} + 2 \sqrt{x} + \frac{x^{2}}{2} + 3 x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x**(1/2))**3/x**(1/2),x)

[Out]

2*x**(3/2) + 2*sqrt(x) + x**2/2 + 3*x

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Giac [B]  time = 1.09194, size = 26, normalized size = 2. \begin{align*} \frac{1}{2} \, x^{2} + 2 \, x^{\frac{3}{2}} + 3 \, x + 2 \, \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x^(1/2))^3/x^(1/2),x, algorithm="giac")

[Out]

1/2*x^2 + 2*x^(3/2) + 3*x + 2*sqrt(x)