3.2391 \(\int \sqrt{3-\frac{1}{\sqrt{x}}} \, dx\)

Optimal. Leaf size=67 \[ \sqrt{3-\frac{1}{\sqrt{x}}} x-\frac{1}{6} \sqrt{3-\frac{1}{\sqrt{x}}} \sqrt{x}-\frac{\tanh ^{-1}\left (\frac{\sqrt{3-\frac{1}{\sqrt{x}}}}{\sqrt{3}}\right )}{6 \sqrt{3}} \]

[Out]

-(Sqrt[3 - 1/Sqrt[x]]*Sqrt[x])/6 + Sqrt[3 - 1/Sqrt[x]]*x - ArcTanh[Sqrt[3 - 1/Sqrt[x]]/Sqrt[3]]/(6*Sqrt[3])

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Rubi [A]  time = 0.0248713, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {190, 47, 51, 63, 206} \[ \sqrt{3-\frac{1}{\sqrt{x}}} x-\frac{1}{6} \sqrt{3-\frac{1}{\sqrt{x}}} \sqrt{x}-\frac{\tanh ^{-1}\left (\frac{\sqrt{3-\frac{1}{\sqrt{x}}}}{\sqrt{3}}\right )}{6 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[3 - 1/Sqrt[x]],x]

[Out]

-(Sqrt[3 - 1/Sqrt[x]]*Sqrt[x])/6 + Sqrt[3 - 1/Sqrt[x]]*x - ArcTanh[Sqrt[3 - 1/Sqrt[x]]/Sqrt[3]]/(6*Sqrt[3])

Rule 190

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(1/n - 1)*(a + b*x)^p, x], x, x^n], x] /
; FreeQ[{a, b, p}, x] && FractionQ[n] && IntegerQ[1/n]

Rule 47

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]

Rule 51

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]

Rule 63

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,
b, c, d, m, n, 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])

Rubi steps

\begin{align*} \int \sqrt{3-\frac{1}{\sqrt{x}}} \, dx &=-\left (2 \operatorname{Subst}\left (\int \frac{\sqrt{3-x}}{x^3} \, dx,x,\frac{1}{\sqrt{x}}\right )\right )\\ &=\sqrt{3-\frac{1}{\sqrt{x}}} x+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{3-x} x^2} \, dx,x,\frac{1}{\sqrt{x}}\right )\\ &=-\frac{1}{6} \sqrt{3-\frac{1}{\sqrt{x}}} \sqrt{x}+\sqrt{3-\frac{1}{\sqrt{x}}} x+\frac{1}{12} \operatorname{Subst}\left (\int \frac{1}{\sqrt{3-x} x} \, dx,x,\frac{1}{\sqrt{x}}\right )\\ &=-\frac{1}{6} \sqrt{3-\frac{1}{\sqrt{x}}} \sqrt{x}+\sqrt{3-\frac{1}{\sqrt{x}}} x-\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{3-x^2} \, dx,x,\sqrt{3-\frac{1}{\sqrt{x}}}\right )\\ &=-\frac{1}{6} \sqrt{3-\frac{1}{\sqrt{x}}} \sqrt{x}+\sqrt{3-\frac{1}{\sqrt{x}}} x-\frac{\tanh ^{-1}\left (\frac{\sqrt{3-\frac{1}{\sqrt{x}}}}{\sqrt{3}}\right )}{6 \sqrt{3}}\\ \end{align*}

Mathematica [C]  time = 0.0135447, size = 36, normalized size = 0.54 \[ \frac{4}{81} \left (3-\frac{1}{\sqrt{x}}\right )^{3/2} \, _2F_1\left (\frac{3}{2},3;\frac{5}{2};1-\frac{1}{3 \sqrt{x}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[3 - 1/Sqrt[x]],x]

[Out]

(4*(3 - 1/Sqrt[x])^(3/2)*Hypergeometric2F1[3/2, 3, 5/2, 1 - 1/(3*Sqrt[x])])/81

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Maple [A]  time = 0.013, size = 91, normalized size = 1.4 \begin{align*} -{\frac{1}{36}\sqrt{{ \left ( 3\,\sqrt{x}-1 \right ){\frac{1}{\sqrt{x}}}}}\sqrt{x} \left ( \ln \left ( -{\frac{\sqrt{3}}{6}}+\sqrt{3}\sqrt{x}+\sqrt{3\,x-\sqrt{x}} \right ) \sqrt{3}-36\,\sqrt{3\,x-\sqrt{x}}\sqrt{x}+6\,\sqrt{3\,x-\sqrt{x}} \right ){\frac{1}{\sqrt{ \left ( 3\,\sqrt{x}-1 \right ) \sqrt{x}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3-1/x^(1/2))^(1/2),x)

[Out]

-1/36*((3*x^(1/2)-1)/x^(1/2))^(1/2)*x^(1/2)*(ln(-1/6*3^(1/2)+3^(1/2)*x^(1/2)+(3*x-x^(1/2))^(1/2))*3^(1/2)-36*(
3*x-x^(1/2))^(1/2)*x^(1/2)+6*(3*x-x^(1/2))^(1/2))/((3*x^(1/2)-1)*x^(1/2))^(1/2)

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Maxima [A]  time = 1.44761, size = 105, normalized size = 1.57 \begin{align*} \frac{1}{36} \, \sqrt{3} \log \left (-\frac{\sqrt{3} - \sqrt{-\frac{1}{\sqrt{x}} + 3}}{\sqrt{3} + \sqrt{-\frac{1}{\sqrt{x}} + 3}}\right ) + \frac{{\left (-\frac{1}{\sqrt{x}} + 3\right )}^{\frac{3}{2}} + 3 \, \sqrt{-\frac{1}{\sqrt{x}} + 3}}{6 \,{\left ({\left (\frac{1}{\sqrt{x}} - 3\right )}^{2} + \frac{6}{\sqrt{x}} - 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3-1/x^(1/2))^(1/2),x, algorithm="maxima")

[Out]

1/36*sqrt(3)*log(-(sqrt(3) - sqrt(-1/sqrt(x) + 3))/(sqrt(3) + sqrt(-1/sqrt(x) + 3))) + 1/6*((-1/sqrt(x) + 3)^(
3/2) + 3*sqrt(-1/sqrt(x) + 3))/((1/sqrt(x) - 3)^2 + 6/sqrt(x) - 9)

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Fricas [A]  time = 1.60011, size = 166, normalized size = 2.48 \begin{align*} \frac{1}{6} \,{\left (6 \, x - \sqrt{x}\right )} \sqrt{\frac{3 \, x - \sqrt{x}}{x}} + \frac{1}{36} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{x} \sqrt{\frac{3 \, x - \sqrt{x}}{x}} - 6 \, \sqrt{x} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3-1/x^(1/2))^(1/2),x, algorithm="fricas")

[Out]

1/6*(6*x - sqrt(x))*sqrt((3*x - sqrt(x))/x) + 1/36*sqrt(3)*log(2*sqrt(3)*sqrt(x)*sqrt((3*x - sqrt(x))/x) - 6*s
qrt(x) + 1)

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Sympy [A]  time = 3.84521, size = 165, normalized size = 2.46 \begin{align*} \begin{cases} \frac{3 x^{\frac{5}{4}}}{\sqrt{3 \sqrt{x} - 1}} - \frac{3 x^{\frac{3}{4}}}{2 \sqrt{3 \sqrt{x} - 1}} + \frac{\sqrt [4]{x}}{6 \sqrt{3 \sqrt{x} - 1}} - \frac{\sqrt{3} \operatorname{acosh}{\left (\sqrt{3} \sqrt [4]{x} \right )}}{18} & \text{for}\: 3 \left |{\sqrt{x}}\right | > 1 \\- \frac{3 i x^{\frac{5}{4}}}{\sqrt{1 - 3 \sqrt{x}}} + \frac{3 i x^{\frac{3}{4}}}{2 \sqrt{1 - 3 \sqrt{x}}} - \frac{i \sqrt [4]{x}}{6 \sqrt{1 - 3 \sqrt{x}}} + \frac{\sqrt{3} i \operatorname{asin}{\left (\sqrt{3} \sqrt [4]{x} \right )}}{18} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3-1/x**(1/2))**(1/2),x)

[Out]

Piecewise((3*x**(5/4)/sqrt(3*sqrt(x) - 1) - 3*x**(3/4)/(2*sqrt(3*sqrt(x) - 1)) + x**(1/4)/(6*sqrt(3*sqrt(x) -
1)) - sqrt(3)*acosh(sqrt(3)*x**(1/4))/18, 3*Abs(sqrt(x)) > 1), (-3*I*x**(5/4)/sqrt(1 - 3*sqrt(x)) + 3*I*x**(3/
4)/(2*sqrt(1 - 3*sqrt(x))) - I*x**(1/4)/(6*sqrt(1 - 3*sqrt(x))) + sqrt(3)*I*asin(sqrt(3)*x**(1/4))/18, True))

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3-1/x^(1/2))^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError