3.835 \(\int \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} \sqrt{x} \, dx\)

Optimal. Leaf size=73 \[ \frac{1}{2} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{3/2}-\frac{1}{4} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}-\frac{1}{4} \cosh ^{-1}\left (\sqrt{x}\right ) \]

[Out]

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

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Rubi [A]  time = 0.107583, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{1}{2} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{3/2}-\frac{1}{4} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}-\frac{1}{4} \cosh ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*Sqrt[x],x]

[Out]

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

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Rubi in Sympy [A]  time = 15.5511, size = 61, normalized size = 0.84 \[ \frac{x^{\frac{3}{2}} \sqrt{\sqrt{x} - 1} \sqrt{\sqrt{x} + 1}}{2} - \frac{\sqrt{x} \sqrt{\sqrt{x} - 1} \sqrt{\sqrt{x} + 1}}{4} - \frac{\operatorname{acosh}{\left (\sqrt{x} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0447519, size = 70, normalized size = 0.96 \[ \frac{1}{4} \left (\sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x} (2 x-1)-\log \left (\sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}+\sqrt{x}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*Sqrt[x],x]

[Out]

(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*Sqrt[x]*(-1 + 2*x) - Log[Sqrt[-1 + Sqrt[x]
]*Sqrt[1 + Sqrt[x]] + Sqrt[x]])/4

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Maple [A]  time = 0.008, size = 52, normalized size = 0.7 \[ -{\frac{1}{4}\sqrt{-1+\sqrt{x}}\sqrt{1+\sqrt{x}} \left ( -2\,{x}^{3/2}\sqrt{-1+x}+\sqrt{x}\sqrt{-1+x}+\ln \left ( \sqrt{x}+\sqrt{-1+x} \right ) \right ){\frac{1}{\sqrt{-1+x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.36469, size = 50, normalized size = 0.68 \[ \frac{1}{2} \,{\left (x - 1\right )}^{\frac{3}{2}} \sqrt{x} + \frac{1}{4} \, \sqrt{x - 1} \sqrt{x} - \frac{1}{4} \, \log \left (2 \, \sqrt{x - 1} + 2 \, \sqrt{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.217703, size = 207, normalized size = 2.84 \[ \frac{128 \, x^{4} - 256 \, x^{3} - 4 \,{\left (32 \, x^{3} - 48 \, x^{2} + 18 \, x - 1\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} + 152 \, x^{2} + 4 \,{\left (4 \,{\left (2 \, x - 1\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - 8 \, x^{2} + 8 \, x - 1\right )} \log \left (2 \, \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - 2 \, x + 1\right ) - 24 \, x - 1}{32 \,{\left (4 \,{\left (2 \, x - 1\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - 8 \, x^{2} + 8 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/32*(128*x^4 - 256*x^3 - 4*(32*x^3 - 48*x^2 + 18*x - 1)*sqrt(x)*sqrt(sqrt(x) +
1)*sqrt(sqrt(x) - 1) + 152*x^2 + 4*(4*(2*x - 1)*sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(s
qrt(x) - 1) - 8*x^2 + 8*x - 1)*log(2*sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1)
- 2*x + 1) - 24*x - 1)/(4*(2*x - 1)*sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1)
- 8*x^2 + 8*x - 1)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(x)*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(sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1),x, algorithm="giac")

[Out]

Exception raised: TypeError