3.9.25 \(\int \frac {1}{(1+\sqrt {x}) \sqrt {-\sqrt {x}+x}} \, dx\)

Optimal. Leaf size=63 \[ 4 \tanh ^{-1}\left (\frac {\sqrt {x-\sqrt {x}}}{\sqrt {x}-1}\right )-2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {x-\sqrt {x}}}{\sqrt {x}-1}\right ) \]

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Rubi [A]  time = 0.04, antiderivative size = 61, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {1397, 843, 620, 206, 724} \begin {gather*} \sqrt {2} \tanh ^{-1}\left (\frac {1-3 \sqrt {x}}{2 \sqrt {2} \sqrt {x-\sqrt {x}}}\right )+4 \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt {x-\sqrt {x}}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((1 + Sqrt[x])*Sqrt[-Sqrt[x] + x]),x]

[Out]

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 620

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2
]], x] /; FreeQ[{b, c}, x]

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 843

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rule 1397

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

Rubi steps

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

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Mathematica [A]  time = 0.06, size = 79, normalized size = 1.25 \begin {gather*} -\sqrt {2} \log \left (\sqrt {x}+1\right )+2 \log \left (-2 \sqrt {x}-2 \sqrt {x-\sqrt {x}}+1\right )+\sqrt {2} \log \left (-3 \sqrt {x}+2 \sqrt {2} \sqrt {x-\sqrt {x}}+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((1 + Sqrt[x])*Sqrt[-Sqrt[x] + x]),x]

[Out]

-(Sqrt[2]*Log[1 + Sqrt[x]]) + 2*Log[1 - 2*Sqrt[x] - 2*Sqrt[-Sqrt[x] + x]] + Sqrt[2]*Log[1 - 3*Sqrt[x] + 2*Sqrt
[2]*Sqrt[-Sqrt[x] + x]]

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IntegrateAlgebraic [A]  time = 0.20, size = 63, normalized size = 1.00 \begin {gather*} 4 \tanh ^{-1}\left (\frac {\sqrt {-\sqrt {x}+x}}{-1+\sqrt {x}}\right )-2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {-\sqrt {x}+x}}{-1+\sqrt {x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((1 + Sqrt[x])*Sqrt[-Sqrt[x] + x]),x]

[Out]

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

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fricas [B]  time = 1.60, size = 102, normalized size = 1.62 \begin {gather*} \frac {1}{2} \, \sqrt {2} \log \left (-\frac {17 \, x^{2} - 4 \, {\left (\sqrt {2} {\left (3 \, x + 5\right )} \sqrt {x} - \sqrt {2} {\left (7 \, x + 1\right )}\right )} \sqrt {x - \sqrt {x}} - 16 \, {\left (3 \, x + 1\right )} \sqrt {x} + 46 \, x + 1}{x^{2} - 2 \, x + 1}\right ) + \log \left (-4 \, \sqrt {x - \sqrt {x}} {\left (2 \, \sqrt {x} - 1\right )} - 8 \, x + 8 \, \sqrt {x} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(2)*log(-(17*x^2 - 4*(sqrt(2)*(3*x + 5)*sqrt(x) - sqrt(2)*(7*x + 1))*sqrt(x - sqrt(x)) - 16*(3*x + 1)*
sqrt(x) + 46*x + 1)/(x^2 - 2*x + 1)) + log(-4*sqrt(x - sqrt(x))*(2*sqrt(x) - 1) - 8*x + 8*sqrt(x) - 1)

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giac [A]  time = 0.86, size = 76, normalized size = 1.21 \begin {gather*} -\sqrt {2} \log \left (\frac {2 \, {\left (\sqrt {2} - \sqrt {x - \sqrt {x}} + \sqrt {x} + 1\right )}}{{\left | 2 \, \sqrt {2} + 2 \, \sqrt {x - \sqrt {x}} - 2 \, \sqrt {x} - 2 \right |}}\right ) - 2 \, \log \left ({\left | 2 \, \sqrt {x - \sqrt {x}} - 2 \, \sqrt {x} + 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(2)*log(2*(sqrt(2) - sqrt(x - sqrt(x)) + sqrt(x) + 1)/abs(2*sqrt(2) + 2*sqrt(x - sqrt(x)) - 2*sqrt(x) - 2
)) - 2*log(abs(2*sqrt(x - sqrt(x)) - 2*sqrt(x) + 1))

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maple [A]  time = 0.09, size = 52, normalized size = 0.83

method	result	size
derivativedivides	\(2 \ln \left (-\frac {1}{2}+\sqrt {x}+\sqrt {-\sqrt {x}+x}\right )+\sqrt {2}\, \arctanh \left (\frac {\left (1-3 \sqrt {x}\right ) \sqrt {2}}{4 \sqrt {\left (1+\sqrt {x}\right )^{2}-3 \sqrt {x}-1}}\right )\)	\(52\)
default	\(-\frac {\sqrt {-\sqrt {x}+x}\, \left (\sqrt {2}\, \arctanh \left (\frac {\left (-1+3 \sqrt {x}\right ) \sqrt {2}}{4 \sqrt {-\sqrt {x}+x}}\right )-2 \ln \left (-\frac {1}{2}+\sqrt {x}+\sqrt {-\sqrt {x}+x}\right )\right )}{\sqrt {\sqrt {x}\, \left (-1+\sqrt {x}\right )}}\)	\(67\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(x - sqrt(x))*(sqrt(x) + 1)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{\sqrt {x-\sqrt {x}}\,\left (\sqrt {x}+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(sqrt(-sqrt(x) + x)*(sqrt(x) + 1)), x)

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