∫ 1_________ (√ ____ −1+x +2√_x )2 √ ___

Optimal. Leaf size=71 \[ -\frac {2 \sqrt {x-1}}{3 (3 x+1)}+\frac {4 \sqrt {x}}{3 (3 x+1)}+\frac {8 \tan ^{-1}\left (\frac {\sqrt {x-1}}{\sqrt {3}}-\frac {\sqrt {x}}{\sqrt {3}}\right )}{3 \sqrt {3}} \]

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Rubi [A] time = 0.22, antiderivative size = 82, normalized size of antiderivative = 1.15, number of steps used = 10, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6742, 51, 63, 203, 47} \begin {gather*} -\frac {2 \sqrt {x-1}}{3 (3 x+1)}+\frac {4 \sqrt {x}}{3 (3 x+1)}+\frac {4 \tan ^{-1}\left (\frac {1}{2} \sqrt {3} \sqrt {x-1}\right )}{3 \sqrt {3}}-\frac {4 \tan ^{-1}\left (\sqrt {3} \sqrt {x}\right )}{3 \sqrt {3}} \end {gather*}

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

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*

(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},

x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] && !(IntegerQ[n] && !IntegerQ[m]) && !(ILeQ[m + n + 2, 0

] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1

))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,

x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[

c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub

st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &

& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

\begin {align*} \int \frac {1}{\left (\sqrt {-1+x}+2 \sqrt {x}\right )^2 \sqrt {-1+x}} \, dx &=\int \left (-\frac {8}{3 \sqrt {-1+x} (1+3 x)^2}-\frac {4 \sqrt {x}}{(1+3 x)^2}+\frac {5}{3 \sqrt {-1+x} (1+3 x)}\right ) \, dx\\ &=\frac {5}{3} \int \frac {1}{\sqrt {-1+x} (1+3 x)} \, dx-\frac {8}{3} \int \frac {1}{\sqrt {-1+x} (1+3 x)^2} \, dx-4 \int \frac {\sqrt {x}}{(1+3 x)^2} \, dx\\ &=-\frac {2 \sqrt {-1+x}}{3 (1+3 x)}+\frac {4 \sqrt {x}}{3 (1+3 x)}-\frac {1}{3} \int \frac {1}{\sqrt {-1+x} (1+3 x)} \, dx-\frac {2}{3} \int \frac {1}{\sqrt {x} (1+3 x)} \, dx+\frac {10}{3} \operatorname {Subst}\left (\int \frac {1}{4+3 x^2} \, dx,x,\sqrt {-1+x}\right )\\ &=-\frac {2 \sqrt {-1+x}}{3 (1+3 x)}+\frac {4 \sqrt {x}}{3 (1+3 x)}+\frac {5 \tan ^{-1}\left (\frac {1}{2} \sqrt {3} \sqrt {-1+x}\right )}{3 \sqrt {3}}-\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{4+3 x^2} \, dx,x,\sqrt {-1+x}\right )-\frac {4}{3} \operatorname {Subst}\left (\int \frac {1}{1+3 x^2} \, dx,x,\sqrt {x}\right )\\ &=-\frac {2 \sqrt {-1+x}}{3 (1+3 x)}+\frac {4 \sqrt {x}}{3 (1+3 x)}+\frac {4 \tan ^{-1}\left (\frac {1}{2} \sqrt {3} \sqrt {-1+x}\right )}{3 \sqrt {3}}-\frac {4 \tan ^{-1}\left (\sqrt {3} \sqrt {x}\right )}{3 \sqrt {3}}\\ \end {align*}

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Mathematica [A] time = 0.10, size = 78, normalized size = 1.10 \begin {gather*} \frac {-6 \sqrt {x-1}+12 \sqrt {x}+4 \sqrt {3} (3 x+1) \tan ^{-1}\left (\frac {1}{2} \sqrt {3} \sqrt {x-1}\right )-4 \sqrt {3} (3 x+1) \tan ^{-1}\left (\sqrt {3} \sqrt {x}\right )}{27 x+9} \end {gather*}

(-6*Sqrt[-1 + x] + 12*Sqrt[x] + 4*Sqrt[3]*(1 + 3*x)*ArcTan[(Sqrt[3]*Sqrt[-1 + x])/2] - 4*Sqrt[3]*(1 + 3*x)*Arc

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IntegrateAlgebraic [A] time = 0.38, size = 71, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt {-1+x}}{3 (1+3 x)}+\frac {4 \sqrt {x}}{3 (1+3 x)}+\frac {8 \tan ^{-1}\left (\frac {\sqrt {-1+x}}{\sqrt {3}}-\frac {\sqrt {x}}{\sqrt {3}}\right )}{3 \sqrt {3}} \end {gather*}

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

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fricas [A] time = 0.45, size = 61, normalized size = 0.86 \begin {gather*} \frac {2 \, {\left (2 \, \sqrt {3} {\left (3 \, x + 1\right )} \arctan \left (\frac {1}{2} \, \sqrt {3} \sqrt {x - 1}\right ) - 2 \, \sqrt {3} {\left (3 \, x + 1\right )} \arctan \left (\sqrt {3} \sqrt {x}\right ) - 3 \, \sqrt {x - 1} + 6 \, \sqrt {x}\right )}}{9 \, {\left (3 \, x + 1\right )}} \end {gather*}

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

2/9*(2*sqrt(3)*(3*x + 1)*arctan(1/2*sqrt(3)*sqrt(x - 1)) - 2*sqrt(3)*(3*x + 1)*arctan(sqrt(3)*sqrt(x)) - 3*sqr

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giac [B] time = 0.62, size = 132, normalized size = 1.86 \begin {gather*} \frac {2}{9} \, \sqrt {3} {\left (\pi - 2 \, \arctan \left (-\frac {\sqrt {3} {\left ({\left (\sqrt {x - 1} - \sqrt {x}\right )}^{2} + 1\right )}}{2 \, {\left (\sqrt {x - 1} - \sqrt {x}\right )}}\right )\right )} + \frac {4}{9} \, \sqrt {3} \arctan \left (\frac {1}{2} \, \sqrt {3} \sqrt {x - 1}\right ) - \frac {8 \, {\left (\sqrt {x - 1} - \sqrt {x} + \frac {1}{\sqrt {x - 1} - \sqrt {x}}\right )}}{3 \, {\left (3 \, {\left (\sqrt {x - 1} - \sqrt {x} + \frac {1}{\sqrt {x - 1} - \sqrt {x}}\right )}^{2} + 4\right )}} - \frac {2 \, \sqrt {x - 1}}{3 \, {\left (3 \, x + 1\right )}} \end {gather*}

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

2/9*sqrt(3)*(pi - 2*arctan(-1/2*sqrt(3)*((sqrt(x - 1) - sqrt(x))^2 + 1)/(sqrt(x - 1) - sqrt(x)))) + 4/9*sqrt(3

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

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

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method	result	size

default	\(-\frac {\sqrt {-1+x}}{4 \left (1+3 x \right )}+\frac {4 \sqrt {3}\, \arctan \left (\frac {\sqrt {-1+x}\, \sqrt {3}}{2}\right )}{9}+\frac {4 \sqrt {x}}{9 \left (\frac {1}{3}+x \right )}-\frac {4 \sqrt {3}\, \arctan \left (\sqrt {x}\, \sqrt {3}\right )}{9}-\frac {5 \sqrt {-1+x}}{36 \left (\frac {1}{3}+x \right )}\)	\(67\)

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

-1/4*(-1+x)^(1/2)/(1+3*x)+4/9*3^(1/2)*arctan(1/2*(-1+x)^(1/2)*3^(1/2))+4/9*x^(1/2)/(1/3+x)-4/9*3^(1/2)*arctan(

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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {x - 1} {\left (\sqrt {x - 1} + 2 \, \sqrt {x}\right )}^{2}}\,{d x} \end {gather*}

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

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mupad [B] time = 1.96, size = 96, normalized size = 1.35 \begin {gather*} \frac {4\,\sqrt {x}}{3\,\left (3\,x+1\right )}-\frac {2\,\sqrt {x-1}}{3\,\left (3\,x+1\right )}+\frac {\sqrt {3}\,\ln \left (\frac {12\,\sqrt {x-1}-\sqrt {3}\,x\,3{}\mathrm {i}+\sqrt {3}\,7{}\mathrm {i}}{3\,x+1}\right )\,2{}\mathrm {i}}{9}+\frac {\sqrt {3}\,\ln \left (\frac {\sqrt {3}-3\,\sqrt {3}\,x+\sqrt {x}\,6{}\mathrm {i}}{x\,3{}\mathrm {i}+1{}\mathrm {i}}\right )\,2{}\mathrm {i}}{9} \end {gather*}

(3^(1/2)*log((12*(x - 1)^(1/2) - 3^(1/2)*x*3i + 3^(1/2)*7i)/(3*x + 1))*2i)/9 - (2*(x - 1)^(1/2))/(3*(3*x + 1))

+ (3^(1/2)*log((3^(1/2) - 3*3^(1/2)*x + x^(1/2)*6i)/(x*3i + 1i))*2i)/9 + (4*x^(1/2))/(3*(3*x + 1))

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sympy [A] time = 20.56, size = 12, normalized size = 0.17 \begin {gather*} \tilde {\infty } \left (- \frac {2 x}{x - 1} + \frac {2}{x - 1}\right ) \end {gather*}

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3.10.31 \(\int \frac {1}{(\sqrt {-1+x}+2 \sqrt {x})^2 \sqrt {-1+x}} \, dx\)