3.2252 \(\int \sqrt{1+\sqrt{x}} \sqrt{x} \, dx\)

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 = 0.035224, 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 \[ \frac{4}{7} \left (\sqrt{x}+1\right )^{7/2}-\frac{8}{5} \left (\sqrt{x}+1\right )^{5/2}+\frac{4}{3} \left (\sqrt{x}+1\right )^{3/2} \]

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

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Rubi in Sympy [A]  time = 4.32721, size = 39, normalized size = 0.85 \[ \frac{4 \left (\sqrt{x} + 1\right )^{\frac{7}{2}}}{7} - \frac{8 \left (\sqrt{x} + 1\right )^{\frac{5}{2}}}{5} + \frac{4 \left (\sqrt{x} + 1\right )^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(1/2)*(1+x**(1/2))**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0134588, size = 34, normalized size = 0.74 \[ \frac{4}{105} \sqrt{\sqrt{x}+1} \left (15 x^{3/2}+3 x-4 \sqrt{x}+8\right ) \]

Antiderivative was successfully verified.

[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

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Maple [A]  time = 0.007, size = 29, normalized size = 0.6 \[{\frac{4}{3} \left ( 1+\sqrt{x} \right ) ^{{\frac{3}{2}}}}-{\frac{8}{5} \left ( 1+\sqrt{x} \right ) ^{{\frac{5}{2}}}}+{\frac{4}{7} \left ( 1+\sqrt{x} \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)*(1+x^(1/2))^(1/2),x)

[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)

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Maxima [A]  time = 1.46878, size = 38, normalized size = 0.83 \[ \frac{4}{7} \,{\left (\sqrt{x} + 1\right )}^{\frac{7}{2}} - \frac{8}{5} \,{\left (\sqrt{x} + 1\right )}^{\frac{5}{2}} + \frac{4}{3} \,{\left (\sqrt{x} + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)*sqrt(sqrt(x) + 1),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 = 0.241923, size = 31, normalized size = 0.67 \[ \frac{4}{105} \,{\left ({\left (15 \, x - 4\right )} \sqrt{x} + 3 \, x + 8\right )} \sqrt{\sqrt{x} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)*sqrt(sqrt(x) + 1),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.35294, size = 398, normalized size = 8.65 \[ \frac{60 x^{\frac{15}{2}} \sqrt{\sqrt{x} + 1}}{315 x^{\frac{11}{2}} + 105 x^{\frac{9}{2}} + 105 x^{6} + 315 x^{5}} + \frac{200 x^{\frac{13}{2}} \sqrt{\sqrt{x} + 1}}{315 x^{\frac{11}{2}} + 105 x^{\frac{9}{2}} + 105 x^{6} + 315 x^{5}} + \frac{60 x^{\frac{11}{2}} \sqrt{\sqrt{x} + 1}}{315 x^{\frac{11}{2}} + 105 x^{\frac{9}{2}} + 105 x^{6} + 315 x^{5}} - \frac{96 x^{\frac{11}{2}}}{315 x^{\frac{11}{2}} + 105 x^{\frac{9}{2}} + 105 x^{6} + 315 x^{5}} + \frac{32 x^{\frac{9}{2}} \sqrt{\sqrt{x} + 1}}{315 x^{\frac{11}{2}} + 105 x^{\frac{9}{2}} + 105 x^{6} + 315 x^{5}} - \frac{32 x^{\frac{9}{2}}}{315 x^{\frac{11}{2}} + 105 x^{\frac{9}{2}} + 105 x^{6} + 315 x^{5}} + \frac{192 x^{7} \sqrt{\sqrt{x} + 1}}{315 x^{\frac{11}{2}} + 105 x^{\frac{9}{2}} + 105 x^{6} + 315 x^{5}} + \frac{80 x^{6} \sqrt{\sqrt{x} + 1}}{315 x^{\frac{11}{2}} + 105 x^{\frac{9}{2}} + 105 x^{6} + 315 x^{5}} - \frac{32 x^{6}}{315 x^{\frac{11}{2}} + 105 x^{\frac{9}{2}} + 105 x^{6} + 315 x^{5}} + \frac{80 x^{5} \sqrt{\sqrt{x} + 1}}{315 x^{\frac{11}{2}} + 105 x^{\frac{9}{2}} + 105 x^{6} + 315 x^{5}} - \frac{96 x^{5}}{315 x^{\frac{11}{2}} + 105 x^{\frac{9}{2}} + 105 x^{6} + 315 x^{5}} \]

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) + 1
92*x**7*sqrt(sqrt(x) + 1)/(315*x**(11/2) + 105*x**(9/2) + 105*x**6 + 315*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**(11/2) + 105*x**(9/2) + 105*x**6 + 315*x**5) + 80*x**5*sqrt(sq
rt(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/XCAS [A]  time = 0.27152, size = 38, normalized size = 0.83 \[ \frac{4}{7} \,{\left (\sqrt{x} + 1\right )}^{\frac{7}{2}} - \frac{8}{5} \,{\left (\sqrt{x} + 1\right )}^{\frac{5}{2}} + \frac{4}{3} \,{\left (\sqrt{x} + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)*sqrt(sqrt(x) + 1),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)