\(\int \sqrt {1+\sqrt {x}} \sqrt {x} \, dx\) [2254]

Optimal result

Integrand size = 17, antiderivative size = 46 \[ \int \sqrt {1+\sqrt {x}} \sqrt {x} \, dx=\frac {4}{3} \left (1+\sqrt {x}\right )^{3/2}-\frac {8}{5} \left (1+\sqrt {x}\right )^{5/2}+\frac {4}{7} \left (1+\sqrt {x}\right )^{7/2} \]

[Out]

4/3*(1+x^(1/2))^(3/2)-8/5*(1+x^(1/2))^(5/2)+4/7*(1+x^(1/2))^(7/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {272, 45} \[ \int \sqrt {1+\sqrt {x}} \sqrt {x} \, dx=\frac {4}{7} \left (\sqrt {x}+1\right )^{7/2}-\frac {8}{5} \left (\sqrt {x}+1\right )^{5/2}+\frac {4}{3} \left (\sqrt {x}+1\right )^{3/2} \]

[In]

Int[Sqrt[1 + Sqrt[x]]*Sqrt[x],x]

[Out]

(4*(1 + Sqrt[x])^(3/2))/3 - (8*(1 + Sqrt[x])^(5/2))/5 + (4*(1 + Sqrt[x])^(7/2))/7

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

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

Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int x^2 \sqrt {1+x} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (\sqrt {1+x}-2 (1+x)^{3/2}+(1+x)^{5/2}\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {4}{3} \left (1+\sqrt {x}\right )^{3/2}-\frac {8}{5} \left (1+\sqrt {x}\right )^{5/2}+\frac {4}{7} \left (1+\sqrt {x}\right )^{7/2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.74 \[ \int \sqrt {1+\sqrt {x}} \sqrt {x} \, dx=\frac {4}{105} \sqrt {1+\sqrt {x}} \left (8-4 \sqrt {x}+3 x+15 x^{3/2}\right ) \]

[In]

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

[Out]

(4*Sqrt[1 + Sqrt[x]]*(8 - 4*Sqrt[x] + 3*x + 15*x^(3/2)))/105

Maple [A] (verified)

Time = 5.95 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.63

method	result	size
derivativedivides	\(\frac {4 \left (\sqrt {x}+1\right )^{\frac {3}{2}}}{3}-\frac {8 \left (\sqrt {x}+1\right )^{\frac {5}{2}}}{5}+\frac {4 \left (\sqrt {x}+1\right )^{\frac {7}{2}}}{7}\)	\(29\)
default	\(\frac {4 \left (\sqrt {x}+1\right )^{\frac {3}{2}}}{3}-\frac {8 \left (\sqrt {x}+1\right )^{\frac {5}{2}}}{5}+\frac {4 \left (\sqrt {x}+1\right )^{\frac {7}{2}}}{7}\)	\(29\)
meijerg	\(-\frac {\frac {32 \sqrt {\pi }}{105}-\frac {4 \sqrt {\pi }\, \left (\sqrt {x}+1\right )^{\frac {3}{2}} \left (15 x -12 \sqrt {x}+8\right )}{105}}{\sqrt {\pi }}\)	\(34\)

[In]

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

[Out]

4/3*(x^(1/2)+1)^(3/2)-8/5*(x^(1/2)+1)^(5/2)+4/7*(x^(1/2)+1)^(7/2)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.50 \[ \int \sqrt {1+\sqrt {x}} \sqrt {x} \, dx=\frac {4}{105} \, {\left ({\left (15 \, x - 4\right )} \sqrt {x} + 3 \, x + 8\right )} \sqrt {\sqrt {x} + 1} \]

[In]

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

[Out]

4/105*((15*x - 4)*sqrt(x) + 3*x + 8)*sqrt(sqrt(x) + 1)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 398 vs. \(2 (39) = 78\).

Time = 0.79 (sec) , antiderivative size = 398, normalized size of antiderivative = 8.65 \[ \int \sqrt {1+\sqrt {x}} \sqrt {x} \, dx=\frac {60 x^{\frac {15}{2}} \sqrt {\sqrt {x} + 1}}{315 x^{\frac {11}{2}} + 105 x^{\frac {9}{2}} + 105 x^{6} + 315 x^{5}} + \frac {200 x^{\frac {13}{2}} \sqrt {\sqrt {x} + 1}}{315 x^{\frac {11}{2}} + 105 x^{\frac {9}{2}} + 105 x^{6} + 315 x^{5}} + \frac {60 x^{\frac {11}{2}} \sqrt {\sqrt {x} + 1}}{315 x^{\frac {11}{2}} + 105 x^{\frac {9}{2}} + 105 x^{6} + 315 x^{5}} - \frac {96 x^{\frac {11}{2}}}{315 x^{\frac {11}{2}} + 105 x^{\frac {9}{2}} + 105 x^{6} + 315 x^{5}} + \frac {32 x^{\frac {9}{2}} \sqrt {\sqrt {x} + 1}}{315 x^{\frac {11}{2}} + 105 x^{\frac {9}{2}} + 105 x^{6} + 315 x^{5}} - \frac {32 x^{\frac {9}{2}}}{315 x^{\frac {11}{2}} + 105 x^{\frac {9}{2}} + 105 x^{6} + 315 x^{5}} + \frac {192 x^{7} \sqrt {\sqrt {x} + 1}}{315 x^{\frac {11}{2}} + 105 x^{\frac {9}{2}} + 105 x^{6} + 315 x^{5}} + \frac {80 x^{6} \sqrt {\sqrt {x} + 1}}{315 x^{\frac {11}{2}} + 105 x^{\frac {9}{2}} + 105 x^{6} + 315 x^{5}} - \frac {32 x^{6}}{315 x^{\frac {11}{2}} + 105 x^{\frac {9}{2}} + 105 x^{6} + 315 x^{5}} + \frac {80 x^{5} \sqrt {\sqrt {x} + 1}}{315 x^{\frac {11}{2}} + 105 x^{\frac {9}{2}} + 105 x^{6} + 315 x^{5}} - \frac {96 x^{5}}{315 x^{\frac {11}{2}} + 105 x^{\frac {9}{2}} + 105 x^{6} + 315 x^{5}} \]

[In]

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

[Out]

60*x**(15/2)*sqrt(sqrt(x) + 1)/(315*x**(11/2) + 105*x**(9/2) + 105*x**6 + 315*x**5) + 200*x**(13/2)*sqrt(sqrt(
x) + 1)/(315*x**(11/2) + 105*x**(9/2) + 105*x**6 + 315*x**5) + 60*x**(11/2)*sqrt(sqrt(x) + 1)/(315*x**(11/2) +
105*x**(9/2) + 105*x**6 + 315*x**5) - 96*x**(11/2)/(315*x**(11/2) + 105*x**(9/2) + 105*x**6 + 315*x**5) + 32*
x**(9/2)*sqrt(sqrt(x) + 1)/(315*x**(11/2) + 105*x**(9/2) + 105*x**6 + 315*x**5) - 32*x**(9/2)/(315*x**(11/2) +
105*x**(9/2) + 105*x**6 + 315*x**5) + 192*x**7*sqrt(sqrt(x) + 1)/(315*x**(11/2) + 105*x**(9/2) + 105*x**6 + 3
15*x**5) + 80*x**6*sqrt(sqrt(x) + 1)/(315*x**(11/2) + 105*x**(9/2) + 105*x**6 + 315*x**5) - 32*x**6/(315*x**(1
1/2) + 105*x**(9/2) + 105*x**6 + 315*x**5) + 80*x**5*sqrt(sqrt(x) + 1)/(315*x**(11/2) + 105*x**(9/2) + 105*x**
6 + 315*x**5) - 96*x**5/(315*x**(11/2) + 105*x**(9/2) + 105*x**6 + 315*x**5)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.61 \[ \int \sqrt {1+\sqrt {x}} \sqrt {x} \, dx=\frac {4}{7} \, {\left (\sqrt {x} + 1\right )}^{\frac {7}{2}} - \frac {8}{5} \, {\left (\sqrt {x} + 1\right )}^{\frac {5}{2}} + \frac {4}{3} \, {\left (\sqrt {x} + 1\right )}^{\frac {3}{2}} \]

[In]

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

[Out]

4/7*(sqrt(x) + 1)^(7/2) - 8/5*(sqrt(x) + 1)^(5/2) + 4/3*(sqrt(x) + 1)^(3/2)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.61 \[ \int \sqrt {1+\sqrt {x}} \sqrt {x} \, dx=\frac {4}{7} \, {\left (\sqrt {x} + 1\right )}^{\frac {7}{2}} - \frac {8}{5} \, {\left (\sqrt {x} + 1\right )}^{\frac {5}{2}} + \frac {4}{3} \, {\left (\sqrt {x} + 1\right )}^{\frac {3}{2}} \]

[In]

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

[Out]

4/7*(sqrt(x) + 1)^(7/2) - 8/5*(sqrt(x) + 1)^(5/2) + 4/3*(sqrt(x) + 1)^(3/2)

Mupad [F(-1)]

Timed out. \[ \int \sqrt {1+\sqrt {x}} \sqrt {x} \, dx=\int \sqrt {x}\,\sqrt {\sqrt {x}+1} \,d x \]

[In]

int(x^(1/2)*(x^(1/2) + 1)^(1/2),x)

[Out]

int(x^(1/2)*(x^(1/2) + 1)^(1/2), x)