3.840 \(\int \frac{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}}{x^{9/2}} \, dx\)

Optimal. Leaf size=94 \[ \frac{16 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{105 x^{3/2}}+\frac{8 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{35 x^{5/2}}+\frac{2 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{7 x^{7/2}} \]

[Out]

(2*(-1 + Sqrt[x])^(3/2)*(1 + Sqrt[x])^(3/2))/(7*x^(7/2)) + (8*(-1 + Sqrt[x])^(3/
2)*(1 + Sqrt[x])^(3/2))/(35*x^(5/2)) + (16*(-1 + Sqrt[x])^(3/2)*(1 + Sqrt[x])^(3
/2))/(105*x^(3/2))

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Rubi [A]  time = 0.101299, 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 \[ \frac{16 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{105 x^{3/2}}+\frac{8 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{35 x^{5/2}}+\frac{2 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{7 x^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/x^(9/2),x]

[Out]

(2*(-1 + Sqrt[x])^(3/2)*(1 + Sqrt[x])^(3/2))/(7*x^(7/2)) + (8*(-1 + Sqrt[x])^(3/
2)*(1 + Sqrt[x])^(3/2))/(35*x^(5/2)) + (16*(-1 + Sqrt[x])^(3/2)*(1 + Sqrt[x])^(3
/2))/(105*x^(3/2))

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Rubi in Sympy [A]  time = 9.61658, size = 85, normalized size = 0.9 \[ \frac{16 \left (\sqrt{x} - 1\right )^{\frac{3}{2}} \left (\sqrt{x} + 1\right )^{\frac{3}{2}}}{105 x^{\frac{3}{2}}} + \frac{8 \left (\sqrt{x} - 1\right )^{\frac{3}{2}} \left (\sqrt{x} + 1\right )^{\frac{3}{2}}}{35 x^{\frac{5}{2}}} + \frac{2 \left (\sqrt{x} - 1\right )^{\frac{3}{2}} \left (\sqrt{x} + 1\right )^{\frac{3}{2}}}{7 x^{\frac{7}{2}}} \]

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**(9/2),x)

[Out]

16*(sqrt(x) - 1)**(3/2)*(sqrt(x) + 1)**(3/2)/(105*x**(3/2)) + 8*(sqrt(x) - 1)**(
3/2)*(sqrt(x) + 1)**(3/2)/(35*x**(5/2)) + 2*(sqrt(x) - 1)**(3/2)*(sqrt(x) + 1)**
(3/2)/(7*x**(7/2))

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Mathematica [A]  time = 0.0224798, size = 46, normalized size = 0.49 \[ \frac{2 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \left (8 x^3+4 x^2+3 x-15\right )}{105 x^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/x^(9/2),x]

[Out]

(2*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*(-15 + 3*x + 4*x^2 + 8*x^3))/(105*x^(7/2
))

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Maple [A]  time = 0.012, size = 33, normalized size = 0.4 \[{\frac{ \left ( -2+2\,x \right ) \left ( 8\,{x}^{2}+12\,x+15 \right ) }{105}\sqrt{-1+\sqrt{x}}\sqrt{1+\sqrt{x}}{x}^{-{\frac{7}{2}}}} \]

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^(9/2),x)

[Out]

2/105*(-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)*(-1+x)*(8*x^2+12*x+15)/x^(7/2)

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Maxima [A]  time = 1.51427, size = 42, normalized size = 0.45 \[ \frac{16 \,{\left (x - 1\right )}^{\frac{3}{2}}}{105 \, x^{\frac{3}{2}}} + \frac{8 \,{\left (x - 1\right )}^{\frac{3}{2}}}{35 \, x^{\frac{5}{2}}} + \frac{2 \,{\left (x - 1\right )}^{\frac{3}{2}}}{7 \, x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1)/x^(9/2),x, algorithm="maxima")

[Out]

16/105*(x - 1)^(3/2)/x^(3/2) + 8/35*(x - 1)^(3/2)/x^(5/2) + 2/7*(x - 1)^(3/2)/x^
(7/2)

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Fricas [A]  time = 0.213274, size = 161, normalized size = 1.71 \[ \frac{2 \,{\left (1120 \, x^{4} - 2380 \, x^{3} - 7 \,{\left (160 \, x^{3} - 260 \, x^{2} + 123 \, x - 15\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} + 1631 \, x^{2} - 378 \, x + 15\right )}}{105 \,{\left (64 \, x^{7} - 112 \, x^{6} + 56 \, x^{5} - 7 \, x^{4} -{\left (64 \, x^{6} - 80 \, x^{5} + 24 \, x^{4} - x^{3}\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1)/x^(9/2),x, algorithm="fricas")

[Out]

2/105*(1120*x^4 - 2380*x^3 - 7*(160*x^3 - 260*x^2 + 123*x - 15)*sqrt(x)*sqrt(sqr
t(x) + 1)*sqrt(sqrt(x) - 1) + 1631*x^2 - 378*x + 15)/(64*x^7 - 112*x^6 + 56*x^5
- 7*x^4 - (64*x^6 - 80*x^5 + 24*x^4 - x^3)*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**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.223258, size = 150, normalized size = 1.6 \[ \frac{4096 \,{\left (35 \,{\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{16} - 70 \,{\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{12} + 168 \,{\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{8} + 224 \,{\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{4} + 128\right )}}{105 \,{\left ({\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{4} + 4\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1)/x^(9/2),x, algorithm="giac")

[Out]

4096/105*(35*(sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^16 - 70*(sqrt(sqrt(x) + 1)
- sqrt(sqrt(x) - 1))^12 + 168*(sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^8 + 224*(s
qrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^4 + 128)/((sqrt(sqrt(x) + 1) - sqrt(sqrt(x
) - 1))^4 + 4)^7