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

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

[Out]

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

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Rubi [A]  time = 0.112956, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107 \[ \frac{1}{2} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{3/2}+\frac{3}{4} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}+\frac{3}{4} \cosh ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 13.133, size = 65, normalized size = 0.89 \[ \frac{x^{\frac{3}{2}} \sqrt{\sqrt{x} - 1} \sqrt{\sqrt{x} + 1}}{2} + \frac{3 \sqrt{x} \sqrt{\sqrt{x} - 1} \sqrt{\sqrt{x} + 1}}{4} + \frac{3 \operatorname{acosh}{\left (\sqrt{x} \right )}}{4} \]

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

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Mathematica [A]  time = 0.0461284, size = 70, normalized size = 0.96 \[ \frac{1}{4} \left (\sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x} (2 x+3)+3 \log \left (\sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}+\sqrt{x}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.014, size = 55, normalized size = 0.8 \[{\frac{1}{4}\sqrt{-1+\sqrt{x}}\sqrt{1+\sqrt{x}} \left ( 2\,{x}^{3/2}\sqrt{-1+x}+3\,\sqrt{x}\sqrt{-1+x}+3\,\ln \left ( \sqrt{x}+\sqrt{-1+x} \right ) \right ){\frac{1}{\sqrt{-1+x}}}} \]

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]

1/4*(-1+x^(1/2))^(1/2)*(1+x^(1/2))^(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 = 1.37467, size = 50, normalized size = 0.68 \[ \frac{1}{2} \, \sqrt{x - 1} x^{\frac{3}{2}} + \frac{3}{4} \, \sqrt{x - 1} \sqrt{x} + \frac{3}{4} \, \log \left (2 \, \sqrt{x - 1} + 2 \, \sqrt{x}\right ) \]

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]

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

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Fricas [A]  time = 0.214932, size = 200, normalized size = 2.74 \[ \frac{128 \, x^{4} - 4 \,{\left (32 \, x^{3} + 16 \, x^{2} - 30 \, x - 1\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - 168 \, x^{2} - 12 \,{\left (4 \,{\left (2 \, x - 1\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - 8 \, x^{2} + 8 \, x - 1\right )} \log \left (2 \, \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - 2 \, x + 1\right ) + 40 \, x + 7}{32 \,{\left (4 \,{\left (2 \, x - 1\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - 8 \, x^{2} + 8 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/32*(128*x^4 - 4*(32*x^3 + 16*x^2 - 30*x - 1)*sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sq
rt(x) - 1) - 168*x^2 - 12*(4*(2*x - 1)*sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) -
1) - 8*x^2 + 8*x - 1)*log(2*sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) - 2*x +
1) + 40*x + 7)/(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. \[ \int \frac{x^{\frac{3}{2}}}{\sqrt{\sqrt{x} - 1} \sqrt{\sqrt{x} + 1}}\, dx \]

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