3.2255 \(\int \frac{\sqrt{1+\sqrt{x}}}{\sqrt{x}} \, dx\)

Optimal. Leaf size=15 \[ \frac{4}{3} \left (\sqrt{x}+1\right )^{3/2} \]

[Out]

(4*(1 + Sqrt[x])^(3/2))/3

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Rubi [A]  time = 0.0023161, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {261} \[ \frac{4}{3} \left (\sqrt{x}+1\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + Sqrt[x]]/Sqrt[x],x]

[Out]

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

Mathematica [A]  time = 0.0022676, size = 15, normalized size = 1. \[ \frac{4}{3} \left (\sqrt{x}+1\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + Sqrt[x]]/Sqrt[x],x]

[Out]

(4*(1 + Sqrt[x])^(3/2))/3

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Maple [A]  time = 0.001, size = 10, normalized size = 0.7 \begin{align*}{\frac{4}{3} \left ( \sqrt{x}+1 \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/3*(x^(1/2)+1)^(3/2)

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Maxima [A]  time = 0.970264, size = 12, normalized size = 0.8 \begin{align*} \frac{4}{3} \,{\left (\sqrt{x} + 1\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/3*(sqrt(x) + 1)^(3/2)

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Fricas [A]  time = 1.22614, size = 34, normalized size = 2.27 \begin{align*} \frac{4}{3} \,{\left (\sqrt{x} + 1\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/3*(sqrt(x) + 1)^(3/2)

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Sympy [B]  time = 0.283027, size = 31, normalized size = 2.07 \begin{align*} \frac{4 \sqrt{x} \sqrt{\sqrt{x} + 1}}{3} + \frac{4 \sqrt{\sqrt{x} + 1}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*sqrt(x)*sqrt(sqrt(x) + 1)/3 + 4*sqrt(sqrt(x) + 1)/3

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Giac [A]  time = 1.11483, size = 12, normalized size = 0.8 \begin{align*} \frac{4}{3} \,{\left (\sqrt{x} + 1\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/3*(sqrt(x) + 1)^(3/2)