3.838 \(\int \frac{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}}{x^{5/2}} \, dx\)

Optimal. Leaf size=31 \[ \frac{2 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{3 x^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0353549, antiderivative size = 31, 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 \[ \frac{2 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/x^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.33842, size = 27, normalized size = 0.87 \[ \frac{2 \left (\sqrt{x} - 1\right )^{\frac{3}{2}} \left (\sqrt{x} + 1\right )^{\frac{3}{2}}}{3 x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-1+x**(1/2))**(1/2)*(1+x**(1/2))**(1/2)/x**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.0186333, size = 34, normalized size = 1.1 \[ \frac{2 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} (x-1)}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/x^(5/2),x]

[Out]

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

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Maple [A]  time = 0.006, size = 23, normalized size = 0.7 \[{\frac{-2+2\,x}{3}\sqrt{-1+\sqrt{x}}\sqrt{1+\sqrt{x}}{x}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)/x^(5/2),x)

[Out]

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

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Maxima [A]  time = 1.52182, size = 14, normalized size = 0.45 \[ \frac{2 \,{\left (x - 1\right )}^{\frac{3}{2}}}{3 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1)/x^(5/2),x, algorithm="maxima")

[Out]

2/3*(x - 1)^(3/2)/x^(3/2)

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Fricas [A]  time = 0.215258, size = 104, normalized size = 3.35 \[ -\frac{2 \,{\left (3 \,{\left (2 \, x - 1\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - 6 \, x^{2} + 6 \, x - 1\right )}}{3 \,{\left (4 \, x^{3} -{\left (4 \, x^{2} - x\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - 3 \, x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1)/x^(5/2),x, algorithm="fricas")

[Out]

-2/3*(3*(2*x - 1)*sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) - 6*x^2 + 6*x - 1)
/(4*x^3 - (4*x^2 - x)*sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) - 3*x^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\sqrt{x} - 1} \sqrt{\sqrt{x} + 1}}{x^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-1+x**(1/2))**(1/2)*(1+x**(1/2))**(1/2)/x**(5/2),x)

[Out]

Integral(sqrt(sqrt(x) - 1)*sqrt(sqrt(x) + 1)/x**(5/2), x)

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GIAC/XCAS [A]  time = 0.213755, size = 65, normalized size = 2.1 \[ \frac{16 \,{\left (3 \,{\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{8} + 16\right )}}{3 \,{\left ({\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{4} + 4\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1)/x^(5/2),x, algorithm="giac")

[Out]

16/3*(3*(sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^8 + 16)/((sqrt(sqrt(x) + 1) - sq
rt(sqrt(x) - 1))^4 + 4)^3