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

Optimal. Leaf size=63 \[ \frac{4 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{15 x^{3/2}}+\frac{2 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{5 x^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.0680236, 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 \[ \frac{4 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{15 x^{3/2}}+\frac{2 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{5 x^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 7.22809, size = 56, normalized size = 0.89 \[ \frac{4 \left (\sqrt{x} - 1\right )^{\frac{3}{2}} \left (\sqrt{x} + 1\right )^{\frac{3}{2}}}{15 x^{\frac{3}{2}}} + \frac{2 \left (\sqrt{x} - 1\right )^{\frac{3}{2}} \left (\sqrt{x} + 1\right )^{\frac{3}{2}}}{5 x^{\frac{5}{2}}} \]

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**(7/2),x)

[Out]

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

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Mathematica [A]  time = 0.0195522, size = 39, normalized size = 0.62 \[ \frac{2 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \left (2 x^2+x-3\right )}{15 x^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 28, normalized size = 0.4 \[{\frac{ \left ( -2+2\,x \right ) \left ( 2\,x+3 \right ) }{15}\sqrt{-1+\sqrt{x}}\sqrt{1+\sqrt{x}}{x}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.52121, size = 28, normalized size = 0.44 \[ \frac{4 \,{\left (x - 1\right )}^{\frac{3}{2}}}{15 \, x^{\frac{3}{2}}} + \frac{2 \,{\left (x - 1\right )}^{\frac{3}{2}}}{5 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.213221, size = 131, normalized size = 2.08 \[ \frac{2 \,{\left (60 \, x^{3} - 5 \,{\left (12 \, x^{2} - 13 \, x + 3\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - 95 \, x^{2} + 40 \, x - 3\right )}}{15 \,{\left (16 \, x^{5} - 20 \, x^{4} + 5 \, x^{3} -{\left (16 \, x^{4} - 12 \, x^{3} + x^{2}\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(60*x^3 - 5*(12*x^2 - 13*x + 3)*sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1)
- 95*x^2 + 40*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))

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.219618, size = 122, normalized size = 1.94 \[ \frac{128 \,{\left (15 \,{\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{12} - 20 \,{\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{8} + 80 \,{\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{4} + 64\right )}}{15 \,{\left ({\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{4} + 4\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

128/15*(15*(sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^12 - 20*(sqrt(sqrt(x) + 1) -
sqrt(sqrt(x) - 1))^8 + 80*(sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^4 + 64)/((sqrt
(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^4 + 4)^5