3.548 \(\int \sqrt{2 x+\sqrt{-1+2 x}} \, dx\)

Optimal. Leaf size=80 \[ \frac{1}{3} \left (2 x+\sqrt{2 x-1}\right )^{3/2}-\frac{1}{8} \left (2 \sqrt{2 x-1}+1\right ) \sqrt{2 x+\sqrt{2 x-1}}-\frac{3}{16} \sinh ^{-1}\left (\frac{2 \sqrt{2 x-1}+1}{\sqrt{3}}\right ) \]

[Out]

(2*x + Sqrt[-1 + 2*x])^(3/2)/3 - (Sqrt[2*x + Sqrt[-1 + 2*x]]*(1 + 2*Sqrt[-1 + 2*
x]))/8 - (3*ArcSinh[(1 + 2*Sqrt[-1 + 2*x])/Sqrt[3]])/16

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Rubi [A]  time = 0.0786265, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \frac{1}{3} \left (2 x+\sqrt{2 x-1}\right )^{3/2}-\frac{1}{8} \left (2 \sqrt{2 x-1}+1\right ) \sqrt{2 x+\sqrt{2 x-1}}-\frac{3}{16} \sinh ^{-1}\left (\frac{2 \sqrt{2 x-1}+1}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[2*x + Sqrt[-1 + 2*x]],x]

[Out]

(2*x + Sqrt[-1 + 2*x])^(3/2)/3 - (Sqrt[2*x + Sqrt[-1 + 2*x]]*(1 + 2*Sqrt[-1 + 2*
x]))/8 - (3*ArcSinh[(1 + 2*Sqrt[-1 + 2*x])/Sqrt[3]])/16

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Rubi in Sympy [A]  time = 2.52634, size = 78, normalized size = 0.98 \[ \frac{\left (2 x + \sqrt{2 x - 1}\right )^{\frac{3}{2}}}{3} - \frac{\sqrt{2 x + \sqrt{2 x - 1}} \left (2 \sqrt{2 x - 1} + 1\right )}{8} - \frac{3 \operatorname{atanh}{\left (\frac{2 \sqrt{2 x - 1} + 1}{2 \sqrt{2 x + \sqrt{2 x - 1}}} \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(2*x + sqrt(2*x - 1))**(3/2)/3 - sqrt(2*x + sqrt(2*x - 1))*(2*sqrt(2*x - 1) + 1)
/8 - 3*atanh((2*sqrt(2*x - 1) + 1)/(2*sqrt(2*x + sqrt(2*x - 1))))/16

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Mathematica [A]  time = 0.055038, size = 62, normalized size = 0.78 \[ \frac{1}{48} \left (2 \sqrt{2 x+\sqrt{2 x-1}} \left (16 x+2 \sqrt{2 x-1}-3\right )-9 \sinh ^{-1}\left (\frac{2 \sqrt{2 x-1}+1}{\sqrt{3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[2*x + Sqrt[-1 + 2*x]],x]

[Out]

(2*Sqrt[2*x + Sqrt[-1 + 2*x]]*(-3 + 16*x + 2*Sqrt[-1 + 2*x]) - 9*ArcSinh[(1 + 2*
Sqrt[-1 + 2*x])/Sqrt[3]])/48

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Maple [A]  time = 0.009, size = 60, normalized size = 0.8 \[{\frac{1}{3} \left ( 2\,x+\sqrt{2\,x-1} \right ) ^{{\frac{3}{2}}}}-{\frac{1}{8} \left ( 1+2\,\sqrt{2\,x-1} \right ) \sqrt{2\,x+\sqrt{2\,x-1}}}-{\frac{3}{16}{\it Arcsinh} \left ({\frac{2\,\sqrt{3}}{3} \left ( \sqrt{2\,x-1}+{\frac{1}{2}} \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(2*x+(2*x-1)^(1/2))^(3/2)-1/8*(1+2*(2*x-1)^(1/2))*(2*x+(2*x-1)^(1/2))^(1/2)-
3/16*arcsinh(2/3*3^(1/2)*((2*x-1)^(1/2)+1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{2 \, x + \sqrt{2 \, x - 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(2*x + sqrt(2*x - 1)), x)

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Fricas [A]  time = 0.574076, size = 99, normalized size = 1.24 \[ \frac{1}{24} \,{\left (16 \, x + 2 \, \sqrt{2 \, x - 1} - 3\right )} \sqrt{2 \, x + \sqrt{2 \, x - 1}} + \frac{3}{32} \, \log \left (-4 \, \sqrt{2 \, x + \sqrt{2 \, x - 1}}{\left (2 \, \sqrt{2 \, x - 1} + 1\right )} + 16 \, x + 8 \, \sqrt{2 \, x - 1} - 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*(16*x + 2*sqrt(2*x - 1) - 3)*sqrt(2*x + sqrt(2*x - 1)) + 3/32*log(-4*sqrt(2
*x + sqrt(2*x - 1))*(2*sqrt(2*x - 1) + 1) + 16*x + 8*sqrt(2*x - 1) - 3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.287032, size = 92, normalized size = 1.15 \[ \frac{1}{24} \,{\left (2 \, \sqrt{2 \, x - 1}{\left (4 \, \sqrt{2 \, x - 1} + 1\right )} + 5\right )} \sqrt{2 \, x + \sqrt{2 \, x - 1}} + \frac{3}{16} \,{\rm ln}\left (2 \, \sqrt{2 \, x + \sqrt{2 \, x - 1}} - 2 \, \sqrt{2 \, x - 1} - 1\right ) - \frac{5}{24} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*(2*sqrt(2*x - 1)*(4*sqrt(2*x - 1) + 1) + 5)*sqrt(2*x + sqrt(2*x - 1)) + 3/1
6*ln(2*sqrt(2*x + sqrt(2*x - 1)) - 2*sqrt(2*x - 1) - 1) - 5/24