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

Optimal. Leaf size=135 \[ \frac{1}{4} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{7/2}-\frac{1}{24} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{5/2}-\frac{5}{96} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{3/2}-\frac{5}{64} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}-\frac{5}{64} \cosh ^{-1}\left (\sqrt{x}\right ) \]

[Out]

(-5*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*Sqrt[x])/64 - (5*Sqrt[-1 + Sqrt[x]]*Sqr
t[1 + Sqrt[x]]*x^(3/2))/96 - (Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(5/2))/24 +
(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(7/2))/4 - (5*ArcCosh[Sqrt[x]])/64

_______________________________________________________________________________________

Rubi [A]  time = 0.196234, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{1}{4} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{7/2}-\frac{1}{24} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{5/2}-\frac{5}{96} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{3/2}-\frac{5}{64} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}-\frac{5}{64} \cosh ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-5*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*Sqrt[x])/64 - (5*Sqrt[-1 + Sqrt[x]]*Sqr
t[1 + Sqrt[x]]*x^(3/2))/96 - (Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(5/2))/24 +
(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(7/2))/4 - (5*ArcCosh[Sqrt[x]])/64

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 19.7215, size = 121, normalized size = 0.9 \[ \frac{x^{\frac{7}{2}} \sqrt{\sqrt{x} - 1} \sqrt{\sqrt{x} + 1}}{4} - \frac{x^{\frac{5}{2}} \sqrt{\sqrt{x} - 1} \sqrt{\sqrt{x} + 1}}{24} - \frac{5 x^{\frac{3}{2}} \sqrt{\sqrt{x} - 1} \sqrt{\sqrt{x} + 1}}{96} - \frac{5 \sqrt{x} \sqrt{\sqrt{x} - 1} \sqrt{\sqrt{x} + 1}}{64} - \frac{5 \operatorname{acosh}{\left (\sqrt{x} \right )}}{64} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**(7/2)*sqrt(sqrt(x) - 1)*sqrt(sqrt(x) + 1)/4 - x**(5/2)*sqrt(sqrt(x) - 1)*sqrt
(sqrt(x) + 1)/24 - 5*x**(3/2)*sqrt(sqrt(x) - 1)*sqrt(sqrt(x) + 1)/96 - 5*sqrt(x)
*sqrt(sqrt(x) - 1)*sqrt(sqrt(x) + 1)/64 - 5*acosh(sqrt(x))/64

_______________________________________________________________________________________

Mathematica [A]  time = 0.0679254, size = 80, normalized size = 0.59 \[ \frac{1}{192} \left (\sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x} \left (48 x^3-8 x^2-10 x-15\right )-15 \log \left (\sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}+\sqrt{x}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*Sqrt[x]*(-15 - 10*x - 8*x^2 + 48*x^3) - 15
*Log[Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]] + Sqrt[x]])/192

_______________________________________________________________________________________

Maple [A]  time = 0.013, size = 75, normalized size = 0.6 \[ -{\frac{1}{192}\sqrt{-1+\sqrt{x}}\sqrt{1+\sqrt{x}} \left ( -48\,{x}^{7/2}\sqrt{-1+x}+8\,{x}^{5/2}\sqrt{-1+x}+10\,{x}^{3/2}\sqrt{-1+x}+15\,\sqrt{x}\sqrt{-1+x}+15\,\ln \left ( \sqrt{x}+\sqrt{-1+x} \right ) \right ){\frac{1}{\sqrt{-1+x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/192*(-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)*(-48*x^(7/2)*(-1+x)^(1/2)+8*x^(5/2)*
(-1+x)^(1/2)+10*x^(3/2)*(-1+x)^(1/2)+15*x^(1/2)*(-1+x)^(1/2)+15*ln(x^(1/2)+(-1+x
)^(1/2)))/(-1+x)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.36581, size = 77, normalized size = 0.57 \[ \frac{1}{4} \,{\left (x - 1\right )}^{\frac{3}{2}} x^{\frac{5}{2}} + \frac{5}{24} \,{\left (x - 1\right )}^{\frac{3}{2}} x^{\frac{3}{2}} + \frac{5}{32} \,{\left (x - 1\right )}^{\frac{3}{2}} \sqrt{x} + \frac{5}{64} \, \sqrt{x - 1} \sqrt{x} - \frac{5}{64} \, \log \left (2 \, \sqrt{x - 1} + 2 \, \sqrt{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(x - 1)^(3/2)*x^(5/2) + 5/24*(x - 1)^(3/2)*x^(3/2) + 5/32*(x - 1)^(3/2)*sqrt
(x) + 5/64*sqrt(x - 1)*sqrt(x) - 5/64*log(2*sqrt(x - 1) + 2*sqrt(x))

_______________________________________________________________________________________

Fricas [A]  time = 0.218427, size = 315, normalized size = 2.33 \[ -\frac{98304 \, x^{8} - 262144 \, x^{7} + 229376 \, x^{6} - 81920 \, x^{5} + 43136 \, x^{4} - 37120 \, x^{3} - 8 \,{\left (12288 \, x^{7} - 26624 \, x^{6} + 16896 \, x^{5} - 4352 \, x^{4} + 4144 \, x^{3} - 2760 \, x^{2} + 350 \, x + 29\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} + 10400 \, x^{2} - 120 \,{\left (128 \, x^{4} - 256 \, x^{3} - 8 \,{\left (16 \, x^{3} - 24 \, x^{2} + 10 \, x - 1\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} + 160 \, x^{2} - 32 \, x + 1\right )} \log \left (2 \, \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - 2 \, x + 1\right ) - 32 \, x - 59}{3072 \,{\left (128 \, x^{4} - 256 \, x^{3} - 8 \,{\left (16 \, x^{3} - 24 \, x^{2} + 10 \, x - 1\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} + 160 \, x^{2} - 32 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3072*(98304*x^8 - 262144*x^7 + 229376*x^6 - 81920*x^5 + 43136*x^4 - 37120*x^3
- 8*(12288*x^7 - 26624*x^6 + 16896*x^5 - 4352*x^4 + 4144*x^3 - 2760*x^2 + 350*x
+ 29)*sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) + 10400*x^2 - 120*(128*x^4 -
256*x^3 - 8*(16*x^3 - 24*x^2 + 10*x - 1)*sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x)
- 1) + 160*x^2 - 32*x + 1)*log(2*sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) - 2
*x + 1) - 32*x - 59)/(128*x^4 - 256*x^3 - 8*(16*x^3 - 24*x^2 + 10*x - 1)*sqrt(x)
*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) + 160*x^2 - 32*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(x**(5/2)*(-1+x**(1/2))**(1/2)*(1+x**(1/2))**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError