3.834 \(\int \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{3/2} \, dx\)

Optimal. Leaf size=104 \[ \frac{1}{3} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{5/2}-\frac{1}{12} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{3/2}-\frac{1}{8} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}-\frac{1}{8} \cosh ^{-1}\left (\sqrt{x}\right ) \]

[Out]

-(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*Sqrt[x])/8 - (Sqrt[-1 + Sqrt[x]]*Sqrt[1 +
Sqrt[x]]*x^(3/2))/12 + (Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(5/2))/3 - ArcCo
sh[Sqrt[x]]/8

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Rubi [A]  time = 0.148321, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{1}{3} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{5/2}-\frac{1}{12} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{3/2}-\frac{1}{8} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}-\frac{1}{8} \cosh ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(3/2),x]

[Out]

-(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*Sqrt[x])/8 - (Sqrt[-1 + Sqrt[x]]*Sqrt[1 +
Sqrt[x]]*x^(3/2))/12 + (Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(5/2))/3 - ArcCo
sh[Sqrt[x]]/8

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Rubi in Sympy [A]  time = 17.0903, size = 88, normalized size = 0.85 \[ \frac{x^{\frac{5}{2}} \sqrt{\sqrt{x} - 1} \sqrt{\sqrt{x} + 1}}{3} - \frac{x^{\frac{3}{2}} \sqrt{\sqrt{x} - 1} \sqrt{\sqrt{x} + 1}}{12} - \frac{\sqrt{x} \sqrt{\sqrt{x} - 1} \sqrt{\sqrt{x} + 1}}{8} - \frac{\operatorname{acosh}{\left (\sqrt{x} \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(3/2)*(-1+x**(1/2))**(1/2)*(1+x**(1/2))**(1/2),x)

[Out]

x**(5/2)*sqrt(sqrt(x) - 1)*sqrt(sqrt(x) + 1)/3 - x**(3/2)*sqrt(sqrt(x) - 1)*sqrt
(sqrt(x) + 1)/12 - sqrt(x)*sqrt(sqrt(x) - 1)*sqrt(sqrt(x) + 1)/8 - acosh(sqrt(x)
)/8

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Mathematica [A]  time = 0.051077, size = 75, normalized size = 0.72 \[ \frac{1}{24} \left (\sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x} \left (8 x^2-2 x-3\right )-3 \log \left (\sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}+\sqrt{x}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(3/2),x]

[Out]

(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*Sqrt[x]*(-3 - 2*x + 8*x^2) - 3*Log[Sqrt[-1
+ Sqrt[x]]*Sqrt[1 + Sqrt[x]] + Sqrt[x]])/24

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Maple [A]  time = 0.009, size = 65, normalized size = 0.6 \[ -{\frac{1}{24}\sqrt{-1+\sqrt{x}}\sqrt{1+\sqrt{x}} \left ( -8\,{x}^{5/2}\sqrt{-1+x}+2\,{x}^{3/2}\sqrt{-1+x}+3\,\sqrt{x}\sqrt{-1+x}+3\,\ln \left ( \sqrt{x}+\sqrt{-1+x} \right ) \right ){\frac{1}{\sqrt{-1+x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3/2)*(-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2),x)

[Out]

-1/24*(-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)*(-8*x^(5/2)*(-1+x)^(1/2)+2*x^(3/2)*(-
1+x)^(1/2)+3*x^(1/2)*(-1+x)^(1/2)+3*ln(x^(1/2)+(-1+x)^(1/2)))/(-1+x)^(1/2)

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Maxima [A]  time = 1.37465, size = 63, normalized size = 0.61 \[ \frac{1}{3} \,{\left (x - 1\right )}^{\frac{3}{2}} x^{\frac{3}{2}} + \frac{1}{4} \,{\left (x - 1\right )}^{\frac{3}{2}} \sqrt{x} + \frac{1}{8} \, \sqrt{x - 1} \sqrt{x} - \frac{1}{8} \, \log \left (2 \, \sqrt{x - 1} + 2 \, \sqrt{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(3/2)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1),x, algorithm="maxima")

[Out]

1/3*(x - 1)^(3/2)*x^(3/2) + 1/4*(x - 1)^(3/2)*sqrt(x) + 1/8*sqrt(x - 1)*sqrt(x)
- 1/8*log(2*sqrt(x - 1) + 2*sqrt(x))

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Fricas [A]  time = 0.21535, size = 261, normalized size = 2.51 \[ -\frac{2048 \, x^{6} - 4608 \, x^{5} + 2688 \, x^{4} + 384 \, x^{3} - 2 \,{\left (1024 \, x^{5} - 1792 \, x^{4} + 576 \, x^{3} + 320 \, x^{2} - 128 \, x - 3\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - 576 \, x^{2} - 12 \,{\left (32 \, x^{3} - 2 \,{\left (16 \, x^{2} - 16 \, x + 3\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - 48 \, x^{2} + 18 \, x - 1\right )} \log \left (2 \, \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - 2 \, x + 1\right ) + 54 \, x + 5}{192 \,{\left (32 \, x^{3} - 2 \,{\left (16 \, x^{2} - 16 \, x + 3\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - 48 \, x^{2} + 18 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(3/2)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1),x, algorithm="fricas")

[Out]

-1/192*(2048*x^6 - 4608*x^5 + 2688*x^4 + 384*x^3 - 2*(1024*x^5 - 1792*x^4 + 576*
x^3 + 320*x^2 - 128*x - 3)*sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) - 576*x^2
- 12*(32*x^3 - 2*(16*x^2 - 16*x + 3)*sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1
) - 48*x^2 + 18*x - 1)*log(2*sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) - 2*x +
1) + 54*x + 5)/(32*x^3 - 2*(16*x^2 - 16*x + 3)*sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(s
qrt(x) - 1) - 48*x^2 + 18*x - 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{\frac{3}{2}} \sqrt{\sqrt{x} - 1} \sqrt{\sqrt{x} + 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(3/2)*(-1+x**(1/2))**(1/2)*(1+x**(1/2))**(1/2),x)

[Out]

Integral(x**(3/2)*sqrt(sqrt(x) - 1)*sqrt(sqrt(x) + 1), x)

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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^(3/2)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1),x, algorithm="giac")

[Out]

Exception raised: TypeError