3.8.1 \(\int \sqrt {1+\sqrt {x}+x} \, dx\) [701]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {1355, 654, 626, 633, 221} \begin {gather*} \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 ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + Sqrt[x] + x],x]

[Out]

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

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt[a])]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 626

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x)*((a + b*x + c*x^2)^p/(2*c*(2*p + 1
))), x] - Dist[p*((b^2 - 4*a*c)/(2*c*(2*p + 1))), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 633

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*(-4*(c/(b^2 - 4*a*c)))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 654

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*((a + b*x + c*x^2)^(p +
1)/(2*c*(p + 1))), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 1355

Int[((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Dist[k, Subst[I
nt[x^(k - 1)*(a + b*x^(k*n) + c*x^(2*k*n))^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && EqQ[n2, 2*n] &
& FractionQ[n]

Rubi steps

\begin {align*} \int \sqrt {1+\sqrt {x}+x} \, dx &=2 \text {Subst}\left (\int x \sqrt {1+x+x^2} \, dx,x,\sqrt {x}\right )\\ &=\frac {2}{3} \left (1+\sqrt {x}+x\right )^{3/2}-\text {Subst}\left (\int \sqrt {1+x+x^2} \, dx,x,\sqrt {x}\right )\\ &=-\frac {1}{4} \left (1+2 \sqrt {x}\right ) \sqrt {1+\sqrt {x}+x}+\frac {2}{3} \left (1+\sqrt {x}+x\right )^{3/2}-\frac {3}{8} \text {Subst}\left (\int \frac {1}{\sqrt {1+x+x^2}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {1}{4} \left (1+2 \sqrt {x}\right ) \sqrt {1+\sqrt {x}+x}+\frac {2}{3} \left (1+\sqrt {x}+x\right )^{3/2}-\frac {1}{8} \sqrt {3} \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{3}}} \, dx,x,1+2 \sqrt {x}\right )\\ &=-\frac {1}{4} \left (1+2 \sqrt {x}\right ) \sqrt {1+\sqrt {x}+x}+\frac {2}{3} \left (1+\sqrt {x}+x\right )^{3/2}-\frac {3}{8} \sinh ^{-1}\left (\frac {1+2 \sqrt {x}}{\sqrt {3}}\right )\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 57, normalized size = 0.92 \begin {gather*} \frac {1}{12} \sqrt {1+\sqrt {x}+x} \left (5+2 \sqrt {x}+8 x\right )+\frac {3}{8} \log \left (-1-2 \sqrt {x}+2 \sqrt {1+\sqrt {x}+x}\right ) \end {gather*}

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*Log[-1 - 2*Sqrt[x] + 2*Sqrt[1 + Sqrt[x] + x]])/8

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Maple [A]
time = 0.02, size = 42, normalized size = 0.68

method	result	size
derivativedivides	\(\frac {2 \left (1+x +\sqrt {x}\right )^{\frac {3}{2}}}{3}-\frac {\left (1+2 \sqrt {x}\right ) \sqrt {1+x +\sqrt {x}}}{4}-\frac {3 \arcsinh \left (\frac {2 \sqrt {3}\, \left (\sqrt {x}+\frac {1}{2}\right )}{3}\right )}{8}\)	\(42\)
default	\(\frac {2 \left (1+x +\sqrt {x}\right )^{\frac {3}{2}}}{3}-\frac {\left (1+2 \sqrt {x}\right ) \sqrt {1+x +\sqrt {x}}}{4}-\frac {3 \arcsinh \left (\frac {2 \sqrt {3}\, \left (\sqrt {x}+\frac {1}{2}\right )}{3}\right )}{8}\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x+x^(1/2))^(1/2),x,method=_RETURNVERBOSE)

[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))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(x + sqrt(x) + 1), x)

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Fricas [A]
time = 0.59, size = 51, normalized size = 0.82 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {\sqrt {x} + x + 1}\, dx \end {gather*}

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)

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Giac [A]
time = 1.74, size = 45, normalized size = 0.73 \begin {gather*} \frac {1}{12} \, {\left (2 \, \sqrt {x} {\left (4 \, \sqrt {x} + 1\right )} + 5\right )} \sqrt {x + \sqrt {x} + 1} + \frac {3}{8} \, \log \left (2 \, \sqrt {x + \sqrt {x} + 1} - 2 \, \sqrt {x} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(2*sqrt(x)*(4*sqrt(x) + 1) + 5)*sqrt(x + sqrt(x) + 1) + 3/8*log(2*sqrt(x + sqrt(x) + 1) - 2*sqrt(x) - 1)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \sqrt {x+\sqrt {x}+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + x^(1/2) + 1)^(1/2),x)

[Out]

int((x + x^(1/2) + 1)^(1/2), x)

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