3.2253 \(\int \frac{\sqrt{1+\sqrt{x}}}{\sqrt{x}} \, dx\)

Optimal. Leaf size=15 \[ \frac{4}{3} \left (\sqrt{x}+1\right )^{3/2} \]

[Out]

(4*(1 + Sqrt[x])^(3/2))/3

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Rubi [A]  time = 0.0115843, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{4}{3} \left (\sqrt{x}+1\right )^{3/2} \]

Antiderivative was successfully verified.

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

[Out]

(4*(1 + Sqrt[x])^(3/2))/3

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Rubi in Sympy [A]  time = 1.68515, size = 12, normalized size = 0.8 \[ \frac{4 \left (\sqrt{x} + 1\right )^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*(sqrt(x) + 1)**(3/2)/3

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Mathematica [A]  time = 0.00515461, size = 15, normalized size = 1. \[ \frac{4}{3} \left (\sqrt{x}+1\right )^{3/2} \]

Antiderivative was successfully verified.

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

[Out]

(4*(1 + Sqrt[x])^(3/2))/3

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Maple [A]  time = 0.003, size = 10, normalized size = 0.7 \[{\frac{4}{3} \left ( 1+\sqrt{x} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.42043, size = 12, normalized size = 0.8 \[ \frac{4}{3} \,{\left (\sqrt{x} + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/3*(sqrt(x) + 1)^(3/2)

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Fricas [A]  time = 0.24265, size = 12, normalized size = 0.8 \[ \frac{4}{3} \,{\left (\sqrt{x} + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/3*(sqrt(x) + 1)^(3/2)

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Sympy [A]  time = 0.500026, size = 31, normalized size = 2.07 \[ \frac{4 \sqrt{x} \sqrt{\sqrt{x} + 1}}{3} + \frac{4 \sqrt{\sqrt{x} + 1}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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GIAC/XCAS [A]  time = 0.223076, size = 12, normalized size = 0.8 \[ \frac{4}{3} \,{\left (\sqrt{x} + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/3*(sqrt(x) + 1)^(3/2)