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

Optimal. Leaf size=68 \[ \frac{2}{3} \left (x+\sqrt{x-1}\right )^{3/2}-\frac{1}{4} \left (2 \sqrt{x-1}+1\right ) \sqrt{x+\sqrt{x-1}}-\frac{3}{8} \sinh ^{-1}\left (\frac{2 \sqrt{x-1}+1}{\sqrt{3}}\right ) \]

[Out]

-((1 + 2*Sqrt[-1 + x])*Sqrt[Sqrt[-1 + x] + x])/4 + (2*(Sqrt[-1 + x] + x)^(3/2))/
3 - (3*ArcSinh[(1 + 2*Sqrt[-1 + x])/Sqrt[3]])/8

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Rubi [A]  time = 0.0772413, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{2}{3} \left (x+\sqrt{x-1}\right )^{3/2}-\frac{1}{4} \left (2 \sqrt{x-1}+1\right ) \sqrt{x+\sqrt{x-1}}-\frac{3}{8} \sinh ^{-1}\left (\frac{2 \sqrt{x-1}+1}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[Sqrt[-1 + x] + x],x]

[Out]

-((1 + 2*Sqrt[-1 + x])*Sqrt[Sqrt[-1 + x] + x])/4 + (2*(Sqrt[-1 + x] + x)^(3/2))/
3 - (3*ArcSinh[(1 + 2*Sqrt[-1 + x])/Sqrt[3]])/8

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Rubi in Sympy [A]  time = 2.52094, size = 66, normalized size = 0.97 \[ \frac{2 \left (x + \sqrt{x - 1}\right )^{\frac{3}{2}}}{3} - \frac{\sqrt{x + \sqrt{x - 1}} \left (2 \sqrt{x - 1} + 1\right )}{4} - \frac{3 \operatorname{atanh}{\left (\frac{2 \sqrt{x - 1} + 1}{2 \sqrt{x + \sqrt{x - 1}}} \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(x + sqrt(x - 1))**(3/2)/3 - sqrt(x + sqrt(x - 1))*(2*sqrt(x - 1) + 1)/4 - 3*a
tanh((2*sqrt(x - 1) + 1)/(2*sqrt(x + sqrt(x - 1))))/8

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Mathematica [A]  time = 0.0365456, size = 54, normalized size = 0.79 \[ \frac{1}{12} \sqrt{x+\sqrt{x-1}} \left (8 x+2 \sqrt{x-1}-3\right )-\frac{3}{8} \sinh ^{-1}\left (\frac{2 \sqrt{x-1}+1}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[Sqrt[-1 + x] + x]*(-3 + 2*Sqrt[-1 + x] + 8*x))/12 - (3*ArcSinh[(1 + 2*Sqrt
[-1 + x])/Sqrt[3]])/8

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Maple [A]  time = 0.008, size = 48, normalized size = 0.7 \[{\frac{2}{3} \left ( x+\sqrt{-1+x} \right ) ^{{\frac{3}{2}}}}-{\frac{1}{4} \left ( 1+2\,\sqrt{-1+x} \right ) \sqrt{x+\sqrt{-1+x}}}-{\frac{3}{8}{\it Arcsinh} \left ({\frac{2\,\sqrt{3}}{3} \left ( \sqrt{-1+x}+{\frac{1}{2}} \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x+(-1+x)^(1/2))^(1/2),x)

[Out]

2/3*(x+(-1+x)^(1/2))^(3/2)-1/4*(1+2*(-1+x)^(1/2))*(x+(-1+x)^(1/2))^(1/2)-3/8*arc
sinh(2/3*3^(1/2)*((-1+x)^(1/2)+1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(x + sqrt(x - 1)), x)

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Fricas [A]  time = 0.606246, size = 80, normalized size = 1.18 \[ \frac{1}{12} \,{\left (8 \, x + 2 \, \sqrt{x - 1} - 3\right )} \sqrt{x + \sqrt{x - 1}} + \frac{3}{16} \, \log \left (-4 \, \sqrt{x + \sqrt{x - 1}}{\left (2 \, \sqrt{x - 1} + 1\right )} + 8 \, x + 8 \, \sqrt{x - 1} - 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(8*x + 2*sqrt(x - 1) - 3)*sqrt(x + sqrt(x - 1)) + 3/16*log(-4*sqrt(x + sqrt
(x - 1))*(2*sqrt(x - 1) + 1) + 8*x + 8*sqrt(x - 1) - 3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(x + sqrt(x - 1)), x)

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GIAC/XCAS [A]  time = 0.3034, size = 72, normalized size = 1.06 \[ \frac{1}{12} \,{\left (2 \, \sqrt{x - 1}{\left (4 \, \sqrt{x - 1} + 1\right )} + 5\right )} \sqrt{x + \sqrt{x - 1}} + \frac{3}{8} \,{\rm ln}\left (2 \, \sqrt{x + \sqrt{x - 1}} - 2 \, \sqrt{x - 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(2*sqrt(x - 1)*(4*sqrt(x - 1) + 1) + 5)*sqrt(x + sqrt(x - 1)) + 3/8*ln(2*sq
rt(x + sqrt(x - 1)) - 2*sqrt(x - 1) - 1)