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

Optimal. Leaf size=77 \[ \frac{2 \sqrt{\sqrt{x}+\sqrt{2 x+2 \sqrt{x}+1}+1} \left (6 x^{3/2}+\sqrt{x}-\left (2-\sqrt{x}\right ) \sqrt{2 x+2 \sqrt{x}+1}+2\right )}{15 \sqrt{x}} \]

[Out]

(2*Sqrt[1 + Sqrt[x] + Sqrt[1 + 2*Sqrt[x] + 2*x]]*(2 + Sqrt[x] + 6*x^(3/2) - (2 - Sqrt[x])*Sqrt[1 + 2*Sqrt[x] +
2*x]))/(15*Sqrt[x])

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Rubi [A]  time = 0.0710915, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.037, Rules used = {2114} \[ \frac{2 \sqrt{\sqrt{x}+\sqrt{2 x+2 \sqrt{x}+1}+1} \left (6 x^{3/2}+\sqrt{x}-\left (2-\sqrt{x}\right ) \sqrt{2 x+2 \sqrt{x}+1}+2\right )}{15 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[1 + Sqrt[x] + Sqrt[1 + 2*Sqrt[x] + 2*x]]*(2 + Sqrt[x] + 6*x^(3/2) - (2 - Sqrt[x])*Sqrt[1 + 2*Sqrt[x] +
2*x]))/(15*Sqrt[x])

Rule 2114

Int[((g_.) + (h_.)*(x_))*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]], x_Symbol] :
> Simp[(2*(f*(5*b*c*g^2 - 2*b^2*g*h - 3*a*c*g*h + 2*a*b*h^2) + c*f*(10*c*g^2 - b*g*h + a*h^2)*x + 9*c^2*f*g*h*
x^2 + 3*c^2*f*h^2*x^3 - (e*g - d*h)*(5*c*g - 2*b*h + c*h*x)*Sqrt[a + b*x + c*x^2])*Sqrt[d + e*x + f*Sqrt[a + b
*x + c*x^2]])/(15*c^2*f*(g + h*x)), x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && EqQ[(e*g - d*h)^2 - f^2*(c*g^2
- b*g*h + a*h^2), 0] && EqQ[2*e^2*g - 2*d*e*h - f^2*(2*c*g - b*h), 0]

Rubi steps

\begin{align*} \int \sqrt{1+\sqrt{x}+\sqrt{1+2 \sqrt{x}+2 x}} \, dx &=2 \operatorname{Subst}\left (\int x \sqrt{1+x+\sqrt{1+2 x+2 x^2}} \, dx,x,\sqrt{x}\right )\\ &=\frac{2 \sqrt{1+\sqrt{x}+\sqrt{1+2 \sqrt{x}+2 x}} \left (2+\sqrt{x}+6 x^{3/2}-\left (2-\sqrt{x}\right ) \sqrt{1+2 \sqrt{x}+2 x}\right )}{15 \sqrt{x}}\\ \end{align*}

Mathematica [A]  time = 0.037998, size = 74, normalized size = 0.96 \[ \frac{2 \sqrt{\sqrt{x}+\sqrt{2 x+2 \sqrt{x}+1}+1} \left (6 x^{3/2}+\sqrt{x}+\left (\sqrt{x}-2\right ) \sqrt{2 x+2 \sqrt{x}+1}+2\right )}{15 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[1 + Sqrt[x] + Sqrt[1 + 2*Sqrt[x] + 2*x]]*(2 + Sqrt[x] + 6*x^(3/2) + (-2 + Sqrt[x])*Sqrt[1 + 2*Sqrt[x]
+ 2*x]))/(15*Sqrt[x])

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Maple [F]  time = 0.014, size = 0, normalized size = 0. \begin{align*} \int \sqrt{1+\sqrt{x}+\sqrt{1+2\,x+2\,\sqrt{x}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x^(1/2)+(1+2*x+2*x^(1/2))^(1/2))^(1/2),x)

[Out]

int((1+x^(1/2)+(1+2*x+2*x^(1/2))^(1/2))^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x^(1/2)+(1+2*x+2*x^(1/2))^(1/2))^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 6.29931, size = 165, normalized size = 2.14 \begin{align*} \frac{2 \,{\left (6 \, x^{2} + \sqrt{2 \, x + 2 \, \sqrt{x} + 1}{\left (x - 2 \, \sqrt{x}\right )} + x + 2 \, \sqrt{x}\right )} \sqrt{\sqrt{2 \, x + 2 \, \sqrt{x} + 1} + \sqrt{x} + 1}}{15 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x^(1/2)+(1+2*x+2*x^(1/2))^(1/2))^(1/2),x, algorithm="fricas")

[Out]

2/15*(6*x^2 + sqrt(2*x + 2*sqrt(x) + 1)*(x - 2*sqrt(x)) + x + 2*sqrt(x))*sqrt(sqrt(2*x + 2*sqrt(x) + 1) + sqrt
(x) + 1)/x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x**(1/2)+(1+2*x+2*x**(1/2))**(1/2))**(1/2),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\sqrt{2 \, x + 2 \, \sqrt{x} + 1} + \sqrt{x} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x^(1/2)+(1+2*x+2*x^(1/2))^(1/2))^(1/2),x, algorithm="giac")

[Out]

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