3.964 \(\int \frac{1}{\sqrt{3+\sqrt{-1+2 x}}} \, dx\)

Optimal. Leaf size=37 \[ \frac{2}{3} \left (\sqrt{2 x-1}+3\right )^{3/2}-6 \sqrt{\sqrt{2 x-1}+3} \]

[Out]

-6*Sqrt[3 + Sqrt[-1 + 2*x]] + (2*(3 + Sqrt[-1 + 2*x])^(3/2))/3

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Rubi [A]  time = 0.0133593, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {247, 190, 43} \[ \frac{2}{3} \left (\sqrt{2 x-1}+3\right )^{3/2}-6 \sqrt{\sqrt{2 x-1}+3} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[3 + Sqrt[-1 + 2*x]],x]

[Out]

-6*Sqrt[3 + Sqrt[-1 + 2*x]] + (2*(3 + Sqrt[-1 + 2*x])^(3/2))/3

Rule 247

Int[((a_.) + (b_.)*(v_)^(n_))^(p_), x_Symbol] :> Dist[1/Coefficient[v, x, 1], Subst[Int[(a + b*x^n)^p, x], x,
v], x] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && NeQ[v, x]

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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0103312, size = 30, normalized size = 0.81 \[ \frac{2}{3} \left (\sqrt{2 x-1}-6\right ) \sqrt{\sqrt{2 x-1}+3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[3 + Sqrt[-1 + 2*x]],x]

[Out]

(2*(-6 + Sqrt[-1 + 2*x])*Sqrt[3 + Sqrt[-1 + 2*x]])/3

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Maple [A]  time = 0.004, size = 28, normalized size = 0.8 \begin{align*}{\frac{2}{3} \left ( 3+\sqrt{2\,x-1} \right ) ^{{\frac{3}{2}}}}-6\,\sqrt{3+\sqrt{2\,x-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.07937, size = 36, normalized size = 0.97 \begin{align*} \frac{2}{3} \,{\left (\sqrt{2 \, x - 1} + 3\right )}^{\frac{3}{2}} - 6 \, \sqrt{\sqrt{2 \, x - 1} + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.43197, size = 66, normalized size = 1.78 \begin{align*} \frac{2}{3} \, \sqrt{\sqrt{2 \, x - 1} + 3}{\left (\sqrt{2 \, x - 1} - 6\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*sqrt(sqrt(2*x - 1) + 3)*(sqrt(2*x - 1) - 6)

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Sympy [B]  time = 0.960737, size = 265, normalized size = 7.16 \begin{align*} - \frac{6 \sqrt{6} \left (x - \frac{1}{2}\right )^{\frac{5}{2}} \sqrt{\sqrt{2} \sqrt{x - \frac{1}{2}} + 3}}{3 \sqrt{6} \left (x - \frac{1}{2}\right )^{\frac{5}{2}} + 9 \sqrt{3} \left (x - \frac{1}{2}\right )^{2}} + \frac{36 \sqrt{2} \left (x - \frac{1}{2}\right )^{\frac{5}{2}}}{3 \sqrt{6} \left (x - \frac{1}{2}\right )^{\frac{5}{2}} + 9 \sqrt{3} \left (x - \frac{1}{2}\right )^{2}} + \frac{4 \sqrt{3} \left (x - \frac{1}{2}\right )^{3} \sqrt{\sqrt{2} \sqrt{x - \frac{1}{2}} + 3}}{3 \sqrt{6} \left (x - \frac{1}{2}\right )^{\frac{5}{2}} + 9 \sqrt{3} \left (x - \frac{1}{2}\right )^{2}} - \frac{36 \sqrt{3} \left (x - \frac{1}{2}\right )^{2} \sqrt{\sqrt{2} \sqrt{x - \frac{1}{2}} + 3}}{3 \sqrt{6} \left (x - \frac{1}{2}\right )^{\frac{5}{2}} + 9 \sqrt{3} \left (x - \frac{1}{2}\right )^{2}} + \frac{108 \left (x - \frac{1}{2}\right )^{2}}{3 \sqrt{6} \left (x - \frac{1}{2}\right )^{\frac{5}{2}} + 9 \sqrt{3} \left (x - \frac{1}{2}\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6*sqrt(6)*(x - 1/2)**(5/2)*sqrt(sqrt(2)*sqrt(x - 1/2) + 3)/(3*sqrt(6)*(x - 1/2)**(5/2) + 9*sqrt(3)*(x - 1/2)*
*2) + 36*sqrt(2)*(x - 1/2)**(5/2)/(3*sqrt(6)*(x - 1/2)**(5/2) + 9*sqrt(3)*(x - 1/2)**2) + 4*sqrt(3)*(x - 1/2)*
*3*sqrt(sqrt(2)*sqrt(x - 1/2) + 3)/(3*sqrt(6)*(x - 1/2)**(5/2) + 9*sqrt(3)*(x - 1/2)**2) - 36*sqrt(3)*(x - 1/2
)**2*sqrt(sqrt(2)*sqrt(x - 1/2) + 3)/(3*sqrt(6)*(x - 1/2)**(5/2) + 9*sqrt(3)*(x - 1/2)**2) + 108*(x - 1/2)**2/
(3*sqrt(6)*(x - 1/2)**(5/2) + 9*sqrt(3)*(x - 1/2)**2)

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Giac [A]  time = 1.20493, size = 43, normalized size = 1.16 \begin{align*} \frac{2}{3} \,{\left (\sqrt{2 \, x - 1} + 3\right )}^{\frac{3}{2}} + 4 \, \sqrt{3} - 6 \, \sqrt{\sqrt{2 \, x - 1} + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(sqrt(2*x - 1) + 3)^(3/2) + 4*sqrt(3) - 6*sqrt(sqrt(2*x - 1) + 3)