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

Optimal. Leaf size=62 \[ \frac{2}{3} \left (x+\sqrt{x}+1\right )^{3/2}-\frac{1}{4} \left (2 \sqrt{x}+1\right ) \sqrt{x+\sqrt{x}+1}-\frac{3}{8} \sinh ^{-1}\left (\frac{2 \sqrt{x}+1}{\sqrt{3}}\right ) \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0524468, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417 \[ \frac{2}{3} \left (x+\sqrt{x}+1\right )^{3/2}-\frac{1}{4} \left (2 \sqrt{x}+1\right ) \sqrt{x+\sqrt{x}+1}-\frac{3}{8} \sinh ^{-1}\left (\frac{2 \sqrt{x}+1}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 + Sqrt[x] + x],x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.05082, size = 63, normalized size = 1.02 \[ - \frac{\left (2 \sqrt{x} + 1\right ) \sqrt{\sqrt{x} + x + 1}}{4} + \frac{2 \left (\sqrt{x} + x + 1\right )^{\frac{3}{2}}}{3} - \frac{3 \operatorname{atanh}{\left (\frac{2 \sqrt{x} + 1}{2 \sqrt{\sqrt{x} + x + 1}} \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*sqrt(x) + 1)*sqrt(sqrt(x) + x + 1)/4 + 2*(sqrt(x) + x + 1)**(3/2)/3 - 3*atan
h((2*sqrt(x) + 1)/(2*sqrt(sqrt(x) + x + 1)))/8

_______________________________________________________________________________________

Mathematica [A]  time = 0.0303024, size = 49, normalized size = 0.79 \[ \frac{1}{12} \sqrt{x+\sqrt{x}+1} \left (8 x+2 \sqrt{x}+5\right )-\frac{3}{8} \sinh ^{-1}\left (\frac{2 \sqrt{x}+1}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 + Sqrt[x] + x]*(5 + 2*Sqrt[x] + 8*x))/12 - (3*ArcSinh[(1 + 2*Sqrt[x])/Sq
rt[3]])/8

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 42, normalized size = 0.7 \[{\frac{2}{3} \left ( 1+x+\sqrt{x} \right ) ^{{\frac{3}{2}}}}-{\frac{1}{4} \left ( 1+2\,\sqrt{x} \right ) \sqrt{1+x+\sqrt{x}}}-{\frac{3}{8}{\it Arcsinh} \left ({\frac{2\,\sqrt{3}}{3} \left ( \sqrt{x}+{\frac{1}{2}} \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x+x^(1/2))^(1/2),x)

[Out]

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

_______________________________________________________________________________________

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)

_______________________________________________________________________________________

Fricas [A]  time = 0.555843, size = 69, normalized size = 1.11 \[ \frac{1}{12} \,{\left (8 \, x + 2 \, \sqrt{x} + 5\right )} \sqrt{x + \sqrt{x} + 1} + \frac{3}{16} \, \log \left (4 \, \sqrt{x + \sqrt{x} + 1}{\left (2 \, \sqrt{x} + 1\right )} - 8 \, x - 8 \, \sqrt{x} - 5\right ) \]

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) + 5)*sqrt(x + sqrt(x) + 1) + 3/16*log(4*sqrt(x + sqrt(x) +
1)*(2*sqrt(x) + 1) - 8*x - 8*sqrt(x) - 5)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(sqrt(x) + x + 1), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.287548, size = 61, normalized size = 0.98 \[ \frac{1}{12} \,{\left (2 \, \sqrt{x}{\left (4 \, \sqrt{x} + 1\right )} + 5\right )} \sqrt{x + \sqrt{x} + 1} + \frac{3}{8} \,{\rm ln}\left (2 \, \sqrt{x + \sqrt{x} + 1} - 2 \, \sqrt{x} - 1\right ) \]

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)*(4*sqrt(x) + 1) + 5)*sqrt(x + sqrt(x) + 1) + 3/8*ln(2*sqrt(x + s
qrt(x) + 1) - 2*sqrt(x) - 1)