3.2254 \(\int \sqrt{1+\sqrt{x}} \sqrt{x} \, dx\)

Optimal. Leaf size=46 \[ \frac{4}{7} \left (\sqrt{x}+1\right )^{7/2}-\frac{8}{5} \left (\sqrt{x}+1\right )^{5/2}+\frac{4}{3} \left (\sqrt{x}+1\right )^{3/2} \]

[Out]

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

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Rubi [A]  time = 0.0120141, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {266, 43} \[ \frac{4}{7} \left (\sqrt{x}+1\right )^{7/2}-\frac{8}{5} \left (\sqrt{x}+1\right )^{5/2}+\frac{4}{3} \left (\sqrt{x}+1\right )^{3/2} \]

Antiderivative was successfully verified.

[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 266

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]]

Rule 43

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])

Rubi steps

\begin{align*} \int \sqrt{1+\sqrt{x}} \sqrt{x} \, dx &=2 \operatorname{Subst}\left (\int x^2 \sqrt{1+x} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{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]  time = 0.0081281, size = 27, normalized size = 0.59 \[ \frac{4}{105} \left (\sqrt{x}+1\right )^{3/2} \left (15 x-12 \sqrt{x}+8\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(1 + Sqrt[x])^(3/2)*(8 - 12*Sqrt[x] + 15*x))/105

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Maple [A]  time = 0.003, size = 29, normalized size = 0.6 \begin{align*}{\frac{4}{3} \left ( \sqrt{x}+1 \right ) ^{{\frac{3}{2}}}}-{\frac{8}{5} \left ( \sqrt{x}+1 \right ) ^{{\frac{5}{2}}}}+{\frac{4}{7} \left ( \sqrt{x}+1 \right ) ^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)

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Maxima [A]  time = 0.99562, size = 38, normalized size = 0.83 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 1.24382, size = 76, normalized size = 1.65 \begin{align*} \frac{4}{105} \,{\left ({\left (15 \, x - 4\right )} \sqrt{x} + 3 \, x + 8\right )} \sqrt{\sqrt{x} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [B]  time = 1.26162, size = 398, normalized size = 8.65 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]  time = 1.08189, size = 38, normalized size = 0.83 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)