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

Optimal. Leaf size=67 \[ \frac{2 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{\sqrt{x}}-2 \sqrt{\sqrt{x}-1} \sqrt{x} \sqrt{\sqrt{x}+1}+2 \cosh ^{-1}\left (\sqrt{x}\right ) \]

[Out]

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

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Rubi [A]  time = 0.106215, antiderivative size = 67, 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 \[ \frac{2 \left (\sqrt{x}-1\right )^{3/2} \left (\sqrt{x}+1\right )^{3/2}}{\sqrt{x}}-2 \sqrt{\sqrt{x}-1} \sqrt{x} \sqrt{\sqrt{x}+1}+2 \cosh ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.2306, size = 61, normalized size = 0.91 \[ - 2 \sqrt{x} \sqrt{\sqrt{x} - 1} \sqrt{\sqrt{x} + 1} + 2 \operatorname{acosh}{\left (\sqrt{x} \right )} + \frac{2 \left (\sqrt{x} - 1\right )^{\frac{3}{2}} \left (\sqrt{x} + 1\right )^{\frac{3}{2}}}{\sqrt{x}} \]

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**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.0268853, size = 62, normalized size = 0.93 \[ 2 \log \left (\sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}+\sqrt{x}\right )-\frac{2 \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{\sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 47, normalized size = 0.7 \[ 2\,{\frac{\sqrt{-1+\sqrt{x}}\sqrt{1+\sqrt{x}} \left ( \ln \left ( \sqrt{x}+\sqrt{-1+x} \right ) \sqrt{x}-\sqrt{-1+x} \right ) }{\sqrt{x}\sqrt{-1+x}}} \]

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^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.53973, size = 36, normalized size = 0.54 \[ -\frac{2 \, \sqrt{x - 1}}{\sqrt{x}} + 2 \, \log \left (2 \, \sqrt{x - 1} + 2 \, \sqrt{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(x - 1)/sqrt(x) + 2*log(2*sqrt(x - 1) + 2*sqrt(x))

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Fricas [A]  time = 0.212356, size = 103, normalized size = 1.54 \[ -\frac{{\left (\sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - x\right )} \log \left (2 \, \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - 2 \, x + 1\right ) - 2}{\sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-((sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) - x)*log(2*sqrt(x)*sqrt(sqrt(x) +
1)*sqrt(sqrt(x) - 1) - 2*x + 1) - 2)/(sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) -
1) - x)

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

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**(3/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.338125, size = 65, normalized size = 0.97 \[ -\frac{16}{{\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{4} + 4} -{\rm ln}\left ({\left (\sqrt{\sqrt{x} + 1} - \sqrt{\sqrt{x} - 1}\right )}^{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-16/((sqrt(sqrt(x) + 1) - sqrt(sqrt(x) - 1))^4 + 4) - ln((sqrt(sqrt(x) + 1) - sq
rt(sqrt(x) - 1))^4)