3.848 \(\int \frac{1}{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{7/2}} \, dx\)

Optimal. Leaf size=94 \[ \frac{8 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{15 x^{3/2}}+\frac{2 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{5 x^{5/2}}+\frac{16 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{15 \sqrt{x}} \]

[Out]

(2*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/(5*x^(5/2)) + (8*Sqrt[-1 + Sqrt[x]]*Sqr
t[1 + Sqrt[x]])/(15*x^(3/2)) + (16*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/(15*Sqr
t[x])

_______________________________________________________________________________________

Rubi [A]  time = 0.112758, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{8 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{15 x^{3/2}}+\frac{2 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{5 x^{5/2}}+\frac{16 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{15 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(7/2)),x]

[Out]

(2*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/(5*x^(5/2)) + (8*Sqrt[-1 + Sqrt[x]]*Sqr
t[1 + Sqrt[x]])/(15*x^(3/2)) + (16*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/(15*Sqr
t[x])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 9.86444, size = 85, normalized size = 0.9 \[ \frac{16 \sqrt{\sqrt{x} - 1} \sqrt{\sqrt{x} + 1}}{15 \sqrt{x}} + \frac{8 \sqrt{\sqrt{x} - 1} \sqrt{\sqrt{x} + 1}}{15 x^{\frac{3}{2}}} + \frac{2 \sqrt{\sqrt{x} - 1} \sqrt{\sqrt{x} + 1}}{5 x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**(7/2)/(-1+x**(1/2))**(1/2)/(1+x**(1/2))**(1/2),x)

[Out]

16*sqrt(sqrt(x) - 1)*sqrt(sqrt(x) + 1)/(15*sqrt(x)) + 8*sqrt(sqrt(x) - 1)*sqrt(s
qrt(x) + 1)/(15*x**(3/2)) + 2*sqrt(sqrt(x) - 1)*sqrt(sqrt(x) + 1)/(5*x**(5/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0277777, size = 41, normalized size = 0.44 \[ \frac{2 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \left (8 x^2+4 x+3\right )}{15 x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(7/2)),x]

[Out]

(2*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*(3 + 4*x + 8*x^2))/(15*x^(5/2))

_______________________________________________________________________________________

Maple [A]  time = 0.013, size = 30, normalized size = 0.3 \[{\frac{16\,{x}^{2}+8\,x+6}{15}\sqrt{-1+\sqrt{x}}\sqrt{1+\sqrt{x}}{x}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(7/2)/(-1+x^(1/2))^(1/2)/(1+x^(1/2))^(1/2),x)

[Out]

2/15*(-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)*(8*x^2+4*x+3)/x^(5/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.52279, size = 42, normalized size = 0.45 \[ \frac{16 \, \sqrt{x - 1}}{15 \, \sqrt{x}} + \frac{8 \, \sqrt{x - 1}}{15 \, x^{\frac{3}{2}}} + \frac{2 \, \sqrt{x - 1}}{5 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^(7/2)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1)),x, algorithm="maxima")

[Out]

16/15*sqrt(x - 1)/sqrt(x) + 8/15*sqrt(x - 1)/x^(3/2) + 2/5*sqrt(x - 1)/x^(5/2)

_______________________________________________________________________________________

Fricas [A]  time = 0.212773, size = 117, normalized size = 1.24 \[ -\frac{2 \,{\left (5 \,{\left (8 \, x - 3\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - 40 \, x^{2} + 35 \, x - 3\right )}}{15 \,{\left (16 \, x^{5} - 20 \, x^{4} + 5 \, x^{3} -{\left (16 \, x^{4} - 12 \, x^{3} + x^{2}\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^(7/2)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1)),x, algorithm="fricas")

[Out]

-2/15*(5*(8*x - 3)*sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) - 40*x^2 + 35*x -
3)/(16*x^5 - 20*x^4 + 5*x^3 - (16*x^4 - 12*x^3 + x^2)*sqrt(x)*sqrt(sqrt(x) + 1)
*sqrt(sqrt(x) - 1))

_______________________________________________________________________________________

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**(7/2)/(-1+x**(1/2))**(1/2)/(1+x**(1/2))**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.220188, size = 93, normalized size = 0.99 \[ \frac{4096 \,{\left (5 \,{\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{8} + 10 \,{\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{4} + 8\right )}}{15 \,{\left ({\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{4} + 4\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^(7/2)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1)),x, algorithm="giac")

[Out]

4096/15*(5*(sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^8 + 10*(sqrt(sqrt(x) + 1) - s
qrt(sqrt(x) - 1))^4 + 8)/((sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^4 + 4)^5