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]

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

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Rubi [A]  time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 15, normalized size = 1.00 \[ \frac {4}{3} \left (\sqrt {x}+1\right )^{3/2} \]

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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fricas [A]  time = 1.06, size = 9, normalized size = 0.60 \[ \frac {4}{3} \, {\left (\sqrt {x} + 1\right )}^{\frac {3}{2}} \]

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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giac [A]  time = 0.19, size = 9, normalized size = 0.60 \[ \frac {4}{3} \, {\left (\sqrt {x} + 1\right )}^{\frac {3}{2}} \]

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)

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maple [A]  time = 0.00, size = 10, normalized size = 0.67 \[ \frac {4 \left (\sqrt {x}+1\right )^{\frac {3}{2}}}{3} \]

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 = 0.88, size = 9, normalized size = 0.60 \[ \frac {4}{3} \, {\left (\sqrt {x} + 1\right )}^{\frac {3}{2}} \]

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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mupad [B]  time = 1.67, size = 9, normalized size = 0.60 \[ \frac {4\,{\left (\sqrt {x}+1\right )}^{3/2}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^(1/2) + 1)^(1/2)/x^(1/2),x)

[Out]

(4*(x^(1/2) + 1)^(3/2))/3

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sympy [B]  time = 0.23, size = 31, normalized size = 2.07 \[ \frac {4 \sqrt {x} \sqrt {\sqrt {x} + 1}}{3} + \frac {4 \sqrt {\sqrt {x} + 1}}{3} \]

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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