3.8.3 \(\int \sqrt {\sqrt {-1+x}+x} \, dx\) [703]

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

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {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}+1\right ) \sqrt {x+\sqrt {x-1}}-\frac {3}{8} \sinh ^{-1}\left (\frac {2 \sqrt {x-1}+1}{\sqrt {3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[Sqrt[-1 + x] + x],x]

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.08, size = 65, normalized size = 0.96 \begin {gather*} \frac {1}{12} \left (5+2 \sqrt {-1+x}+8 (-1+x)\right ) \sqrt {\sqrt {-1+x}+x}+\frac {3}{8} \log \left (-1-2 \sqrt {-1+x}+2 \sqrt {\sqrt {-1+x}+x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[Sqrt[-1 + x] + x],x]

[Out]

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

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Maple [A]
time = 0.02, size = 48, normalized size = 0.71

method	result	size
derivativedivides	\(\frac {2 \left (x +\sqrt {-1+x}\right )^{\frac {3}{2}}}{3}-\frac {\left (1+2 \sqrt {-1+x}\right ) \sqrt {x +\sqrt {-1+x}}}{4}-\frac {3 \arcsinh \left (\frac {2 \sqrt {3}\, \left (\sqrt {-1+x}+\frac {1}{2}\right )}{3}\right )}{8}\)	\(48\)
default	\(\frac {2 \left (x +\sqrt {-1+x}\right )^{\frac {3}{2}}}{3}-\frac {\left (1+2 \sqrt {-1+x}\right ) \sqrt {x +\sqrt {-1+x}}}{4}-\frac {3 \arcsinh \left (\frac {2 \sqrt {3}\, \left (\sqrt {-1+x}+\frac {1}{2}\right )}{3}\right )}{8}\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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*arcsinh(2/3*3^(1/2)*((-1+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((x+(-1+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 = 59, normalized size = 0.87 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x+(-1+x)^(1/2))^(1/2),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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {x + \sqrt {x - 1}}\, dx \end {gather*}

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 [A]
time = 1.51, size = 53, normalized size = 0.78 \begin {gather*} \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} \, \log \left (2 \, \sqrt {x + \sqrt {x - 1}} - 2 \, \sqrt {x - 1} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \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)^(1/2))^(1/2),x)

[Out]

int((x + (x - 1)^(1/2))^(1/2), x)

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