3.846 \(\int \frac{1}{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{3/2}} \, dx\)

Optimal. Leaf size=29 \[ \frac{2 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{\sqrt{x}} \]

[Out]

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

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Rubi [A]  time = 0.0384616, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036 \[ \frac{2 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(3/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.44281, size = 26, normalized size = 0.9 \[ \frac{2 \sqrt{\sqrt{x} - 1} \sqrt{\sqrt{x} + 1}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**(3/2)/(-1+x**(1/2))**(1/2)/(1+x**(1/2))**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0205941, size = 29, normalized size = 1. \[ \frac{2 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(3/2)),x]

[Out]

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

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Maple [A]  time = 0.012, size = 20, normalized size = 0.7 \[ 2\,{\frac{\sqrt{-1+\sqrt{x}}\sqrt{1+\sqrt{x}}}{\sqrt{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(3/2)/(-1+x^(1/2))^(1/2)/(1+x^(1/2))^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.53163, size = 14, normalized size = 0.48 \[ \frac{2 \, \sqrt{x - 1}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^(3/2)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1)),x, algorithm="maxima")

[Out]

2*sqrt(x - 1)/sqrt(x)

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Fricas [A]  time = 0.212085, size = 35, normalized size = 1.21 \[ -\frac{2}{\sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^(3/2)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{\frac{3}{2}} \sqrt{\sqrt{x} - 1} \sqrt{\sqrt{x} + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**(3/2)/(-1+x**(1/2))**(1/2)/(1+x**(1/2))**(1/2),x)

[Out]

Integral(1/(x**(3/2)*sqrt(sqrt(x) - 1)*sqrt(sqrt(x) + 1)), x)

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GIAC/XCAS [A]  time = 0.208807, size = 34, normalized size = 1.17 \[ \frac{16}{{\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{4} + 4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^(3/2)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1)),x, algorithm="giac")

[Out]

16/((sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^4 + 4)