\(\int \frac {\sqrt {1+\sqrt {x}}}{\sqrt {x}} \, dx\) [2255]

Optimal result

Integrand size = 17, antiderivative size = 15 \[ \int \frac {\sqrt {1+\sqrt {x}}}{\sqrt {x}} \, dx=\frac {4}{3} \left (1+\sqrt {x}\right )^{3/2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {267} \[ \int \frac {\sqrt {1+\sqrt {x}}}{\sqrt {x}} \, dx=\frac {4}{3} \left (\sqrt {x}+1\right )^{3/2} \]

[In]

Int[Sqrt[1 + Sqrt[x]]/Sqrt[x],x]

[Out]

(4*(1 + Sqrt[x])^(3/2))/3

Rule 267

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1+\sqrt {x}}}{\sqrt {x}} \, dx=\frac {4}{3} \left (1+\sqrt {x}\right )^{3/2} \]

[In]

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

[Out]

(4*(1 + Sqrt[x])^(3/2))/3

Maple [A] (verified)

Time = 6.01 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.67

method	result	size
derivativedivides	\(\frac {4 \left (\sqrt {x}+1\right )^{\frac {3}{2}}}{3}\)	\(10\)
default	\(\frac {4 \left (\sqrt {x}+1\right )^{\frac {3}{2}}}{3}\)	\(10\)
meijerg	\(-\frac {\frac {4 \sqrt {\pi }}{3}-\frac {2 \sqrt {\pi }\, \left (2 \sqrt {x}+2\right ) \sqrt {\sqrt {x}+1}}{3}}{\sqrt {\pi }}\)	\(31\)

[In]

int((x^(1/2)+1)^(1/2)/x^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.60 \[ \int \frac {\sqrt {1+\sqrt {x}}}{\sqrt {x}} \, dx=\frac {4}{3} \, {\left (\sqrt {x} + 1\right )}^{\frac {3}{2}} \]

[In]

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

[Out]

4/3*(sqrt(x) + 1)^(3/2)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (12) = 24\).

Time = 0.09 (sec) , antiderivative size = 31, normalized size of antiderivative = 2.07 \[ \int \frac {\sqrt {1+\sqrt {x}}}{\sqrt {x}} \, dx=\frac {4 \sqrt {x} \sqrt {\sqrt {x} + 1}}{3} + \frac {4 \sqrt {\sqrt {x} + 1}}{3} \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.60 \[ \int \frac {\sqrt {1+\sqrt {x}}}{\sqrt {x}} \, dx=\frac {4}{3} \, {\left (\sqrt {x} + 1\right )}^{\frac {3}{2}} \]

[In]

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

[Out]

4/3*(sqrt(x) + 1)^(3/2)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.60 \[ \int \frac {\sqrt {1+\sqrt {x}}}{\sqrt {x}} \, dx=\frac {4}{3} \, {\left (\sqrt {x} + 1\right )}^{\frac {3}{2}} \]

[In]

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

[Out]

4/3*(sqrt(x) + 1)^(3/2)

Mupad [B] (verification not implemented)

Time = 6.15 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.60 \[ \int \frac {\sqrt {1+\sqrt {x}}}{\sqrt {x}} \, dx=\frac {4\,{\left (\sqrt {x}+1\right )}^{3/2}}{3} \]

[In]

int((x^(1/2) + 1)^(1/2)/x^(1/2),x)

[Out]

(4*(x^(1/2) + 1)^(3/2))/3