3.841 \(\int \frac{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}}{x^{11/2}} \, dx\)

Optimal. Leaf size=125 \[ \frac{32 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{315 x^{3/2}}+\frac{16 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{105 x^{5/2}}+\frac{4 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{21 x^{7/2}}+\frac{2 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{9 x^{9/2}} \]

[Out]

(2*(-1 + Sqrt[x])^(3/2)*(1 + Sqrt[x])^(3/2))/(9*x^(9/2)) + (4*(-1 + Sqrt[x])^(3/
2)*(1 + Sqrt[x])^(3/2))/(21*x^(7/2)) + (16*(-1 + Sqrt[x])^(3/2)*(1 + Sqrt[x])^(3
/2))/(105*x^(5/2)) + (32*(-1 + Sqrt[x])^(3/2)*(1 + Sqrt[x])^(3/2))/(315*x^(3/2))

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Rubi [A]  time = 0.136567, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{32 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{315 x^{3/2}}+\frac{16 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{105 x^{5/2}}+\frac{4 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{21 x^{7/2}}+\frac{2 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{9 x^{9/2}} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/x^(11/2),x]

[Out]

(2*(-1 + Sqrt[x])^(3/2)*(1 + Sqrt[x])^(3/2))/(9*x^(9/2)) + (4*(-1 + Sqrt[x])^(3/
2)*(1 + Sqrt[x])^(3/2))/(21*x^(7/2)) + (16*(-1 + Sqrt[x])^(3/2)*(1 + Sqrt[x])^(3
/2))/(105*x^(5/2)) + (32*(-1 + Sqrt[x])^(3/2)*(1 + Sqrt[x])^(3/2))/(315*x^(3/2))

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Rubi in Sympy [A]  time = 12.3617, size = 114, normalized size = 0.91 \[ \frac{32 \left (\sqrt{x} - 1\right )^{\frac{3}{2}} \left (\sqrt{x} + 1\right )^{\frac{3}{2}}}{315 x^{\frac{3}{2}}} + \frac{16 \left (\sqrt{x} - 1\right )^{\frac{3}{2}} \left (\sqrt{x} + 1\right )^{\frac{3}{2}}}{105 x^{\frac{5}{2}}} + \frac{4 \left (\sqrt{x} - 1\right )^{\frac{3}{2}} \left (\sqrt{x} + 1\right )^{\frac{3}{2}}}{21 x^{\frac{7}{2}}} + \frac{2 \left (\sqrt{x} - 1\right )^{\frac{3}{2}} \left (\sqrt{x} + 1\right )^{\frac{3}{2}}}{9 x^{\frac{9}{2}}} \]

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**(11/2),x)

[Out]

32*(sqrt(x) - 1)**(3/2)*(sqrt(x) + 1)**(3/2)/(315*x**(3/2)) + 16*(sqrt(x) - 1)**
(3/2)*(sqrt(x) + 1)**(3/2)/(105*x**(5/2)) + 4*(sqrt(x) - 1)**(3/2)*(sqrt(x) + 1)
**(3/2)/(21*x**(7/2)) + 2*(sqrt(x) - 1)**(3/2)*(sqrt(x) + 1)**(3/2)/(9*x**(9/2))

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Mathematica [A]  time = 0.0242314, size = 51, normalized size = 0.41 \[ \frac{2 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \left (16 x^4+8 x^3+6 x^2+5 x-35\right )}{315 x^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]])/x^(11/2),x]

[Out]

(2*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*(-35 + 5*x + 6*x^2 + 8*x^3 + 16*x^4))/(3
15*x^(9/2))

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Maple [A]  time = 0.019, size = 38, normalized size = 0.3 \[{\frac{ \left ( -2+2\,x \right ) \left ( 16\,{x}^{3}+24\,{x}^{2}+30\,x+35 \right ) }{315}\sqrt{-1+\sqrt{x}}\sqrt{1+\sqrt{x}}{x}^{-{\frac{9}{2}}}} \]

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^(11/2),x)

[Out]

2/315*(-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)*(-1+x)*(16*x^3+24*x^2+30*x+35)/x^(9/2
)

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Maxima [A]  time = 1.58064, size = 55, normalized size = 0.44 \[ \frac{32 \,{\left (x - 1\right )}^{\frac{3}{2}}}{315 \, x^{\frac{3}{2}}} + \frac{16 \,{\left (x - 1\right )}^{\frac{3}{2}}}{105 \, x^{\frac{5}{2}}} + \frac{4 \,{\left (x - 1\right )}^{\frac{3}{2}}}{21 \, x^{\frac{7}{2}}} + \frac{2 \,{\left (x - 1\right )}^{\frac{3}{2}}}{9 \, x^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1)/x^(11/2),x, algorithm="maxima")

[Out]

32/315*(x - 1)^(3/2)/x^(3/2) + 16/105*(x - 1)^(3/2)/x^(5/2) + 4/21*(x - 1)^(3/2)
/x^(7/2) + 2/9*(x - 1)^(3/2)/x^(9/2)

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Fricas [A]  time = 0.21033, size = 185, normalized size = 1.48 \[ \frac{2 \,{\left (10080 \, x^{5} - 26712 \, x^{4} + 25242 \, x^{3} - 3 \,{\left (3360 \, x^{4} - 7224 \, x^{3} + 5222 \, x^{2} - 1415 \, x + 105\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - 9999 \, x^{2} + 1440 \, x - 35\right )}}{315 \,{\left (256 \, x^{9} - 576 \, x^{8} + 432 \, x^{7} - 120 \, x^{6} + 9 \, x^{5} -{\left (256 \, x^{8} - 448 \, x^{7} + 240 \, x^{6} - 40 \, x^{5} + x^{4}\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1)/x^(11/2),x, algorithm="fricas")

[Out]

2/315*(10080*x^5 - 26712*x^4 + 25242*x^3 - 3*(3360*x^4 - 7224*x^3 + 5222*x^2 - 1
415*x + 105)*sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) - 9999*x^2 + 1440*x - 3
5)/(256*x^9 - 576*x^8 + 432*x^7 - 120*x^6 + 9*x^5 - (256*x^8 - 448*x^7 + 240*x^6
- 40*x^5 + x^4)*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**(11/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.228976, size = 178, normalized size = 1.42 \[ \frac{16384 \,{\left (315 \,{\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{20} - 756 \,{\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{16} + 1344 \,{\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{12} + 2304 \,{\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{8} + 2304 \,{\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{4} + 1024\right )}}{315 \,{\left ({\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{4} + 4\right )}^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1)/x^(11/2),x, algorithm="giac")

[Out]

16384/315*(315*(sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^20 - 756*(sqrt(sqrt(x) +
1) - sqrt(sqrt(x) - 1))^16 + 1344*(sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^12 + 2
304*(sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^8 + 2304*(sqrt(sqrt(x) + 1) - sqrt(s
qrt(x) - 1))^4 + 1024)/((sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^4 + 4)^9