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

Optimal. Leaf size=63 \[ \frac{2 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{3 x^{3/2}}+\frac{4 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{3 \sqrt{x}} \]

[Out]

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

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Rubi [A]  time = 0.0746418, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{2 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{3 x^{3/2}}+\frac{4 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{3 \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 7.07239, size = 56, normalized size = 0.89 \[ \frac{4 \sqrt{\sqrt{x} - 1} \sqrt{\sqrt{x} + 1}}{3 \sqrt{x}} + \frac{2 \sqrt{\sqrt{x} - 1} \sqrt{\sqrt{x} + 1}}{3 x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.024967, size = 36, normalized size = 0.57 \[ \frac{2 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} (2 x+1)}{3 x^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 25, normalized size = 0.4 \[{\frac{2+4\,x}{3}\sqrt{-1+\sqrt{x}}\sqrt{1+\sqrt{x}}{x}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.52125, size = 28, normalized size = 0.44 \[ \frac{4 \, \sqrt{x - 1}}{3 \, \sqrt{x}} + \frac{2 \, \sqrt{x - 1}}{3 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.211382, size = 90, normalized size = 1.43 \[ -\frac{2 \,{\left (3 \, \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - 3 \, x + 1\right )}}{3 \,{\left (4 \, x^{3} -{\left (4 \, x^{2} - x\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - 3 \, x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(3*sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) - 3*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(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.213209, size = 65, normalized size = 1.03 \[ \frac{128 \,{\left (3 \,{\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{4} + 4\right )}}{3 \,{\left ({\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{4} + 4\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

128/3*(3*(sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^4 + 4)/((sqrt(sqrt(x) + 1) - sq
rt(sqrt(x) - 1))^4 + 4)^3