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

Optimal. Leaf size=29 \[ \frac{4}{3} \left (\sqrt{x}+1\right )^{3/2}-4 \sqrt{\sqrt{x}+1} \]

[Out]

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

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Rubi [A]  time = 0.0177053, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{4}{3} \left (\sqrt{x}+1\right )^{3/2}-4 \sqrt{\sqrt{x}+1} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[1 + Sqrt[x]],x]

[Out]

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

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Rubi in Sympy [A]  time = 1.05162, size = 24, normalized size = 0.83 \[ \frac{4 \left (\sqrt{x} + 1\right )^{\frac{3}{2}}}{3} - 4 \sqrt{\sqrt{x} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00751384, size = 22, normalized size = 0.76 \[ \frac{4}{3} \left (\sqrt{x}-2\right ) \sqrt{\sqrt{x}+1} \]

Antiderivative was successfully verified.

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

[Out]

(4*(-2 + Sqrt[x])*Sqrt[1 + Sqrt[x]])/3

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Maple [A]  time = 0.007, size = 20, normalized size = 0.7 \[{\frac{4}{3} \left ( 1+\sqrt{x} \right ) ^{{\frac{3}{2}}}}-4\,\sqrt{1+\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1+x^(1/2))^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.33752, size = 26, normalized size = 0.9 \[ \frac{4}{3} \,{\left (\sqrt{x} + 1\right )}^{\frac{3}{2}} - 4 \, \sqrt{\sqrt{x} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(sqrt(x) + 1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.204602, size = 19, normalized size = 0.66 \[ \frac{4}{3} \, \sqrt{\sqrt{x} + 1}{\left (\sqrt{x} - 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/3*sqrt(sqrt(x) + 1)*(sqrt(x) - 2)

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Sympy [A]  time = 1.43966, size = 117, normalized size = 4.03 \[ - \frac{4 x^{\frac{5}{2}} \sqrt{\sqrt{x} + 1}}{3 x^{\frac{5}{2}} + 3 x^{2}} + \frac{8 x^{\frac{5}{2}}}{3 x^{\frac{5}{2}} + 3 x^{2}} + \frac{4 x^{3} \sqrt{\sqrt{x} + 1}}{3 x^{\frac{5}{2}} + 3 x^{2}} - \frac{8 x^{2} \sqrt{\sqrt{x} + 1}}{3 x^{\frac{5}{2}} + 3 x^{2}} + \frac{8 x^{2}}{3 x^{\frac{5}{2}} + 3 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.206516, size = 26, normalized size = 0.9 \[ \frac{4}{3} \,{\left (\sqrt{x} + 1\right )}^{\frac{3}{2}} - 4 \, \sqrt{\sqrt{x} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(sqrt(x) + 1),x, algorithm="giac")

[Out]

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