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

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 = 0.0313455, 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 \[ \frac{2}{3} \left (\sqrt{2 x-1}+3\right )^{3/2}-6 \sqrt{\sqrt{2 x-1}+3} \]

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

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Rubi in Sympy [A]  time = 1.37624, size = 31, normalized size = 0.84 \[ \frac{2 \left (\sqrt{2 x - 1} + 3\right )^{\frac{3}{2}}}{3} - 6 \sqrt{\sqrt{2 x - 1} + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(3+(-1+2*x)**(1/2))**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0140015, 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 = 0.006, size = 28, normalized size = 0.8 \[{\frac{2}{3} \left ( 3+\sqrt{2\,x-1} \right ) ^{{\frac{3}{2}}}}-6\,\sqrt{3+\sqrt{2\,x-1}} \]

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 = 0.719549, size = 36, normalized size = 0.97 \[ \frac{2}{3} \,{\left (\sqrt{2 \, x - 1} + 3\right )}^{\frac{3}{2}} - 6 \, \sqrt{\sqrt{2 \, x - 1} + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(sqrt(2*x - 1) + 3),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 = 0.265742, size = 30, normalized size = 0.81 \[ \frac{2}{3} \, \sqrt{\sqrt{2 \, x - 1} + 3}{\left (\sqrt{2 \, x - 1} - 6\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(sqrt(2*x - 1) + 3),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 3.49664, size = 265, normalized size = 7.16 \[ - \frac{6 \sqrt{6} \left (x - \frac{1}{2}\right )^{\frac{5}{2}} \sqrt{\sqrt{2} \sqrt{x - \frac{1}{2}} + 3}}{3 \sqrt{6} \left (x - \frac{1}{2}\right )^{\frac{5}{2}} + 9 \sqrt{3} \left (x - \frac{1}{2}\right )^{2}} + \frac{36 \sqrt{2} \left (x - \frac{1}{2}\right )^{\frac{5}{2}}}{3 \sqrt{6} \left (x - \frac{1}{2}\right )^{\frac{5}{2}} + 9 \sqrt{3} \left (x - \frac{1}{2}\right )^{2}} + \frac{4 \sqrt{3} \left (x - \frac{1}{2}\right )^{3} \sqrt{\sqrt{2} \sqrt{x - \frac{1}{2}} + 3}}{3 \sqrt{6} \left (x - \frac{1}{2}\right )^{\frac{5}{2}} + 9 \sqrt{3} \left (x - \frac{1}{2}\right )^{2}} - \frac{36 \sqrt{3} \left (x - \frac{1}{2}\right )^{2} \sqrt{\sqrt{2} \sqrt{x - \frac{1}{2}} + 3}}{3 \sqrt{6} \left (x - \frac{1}{2}\right )^{\frac{5}{2}} + 9 \sqrt{3} \left (x - \frac{1}{2}\right )^{2}} + \frac{108 \left (x - \frac{1}{2}\right )^{2}}{3 \sqrt{6} \left (x - \frac{1}{2}\right )^{\frac{5}{2}} + 9 \sqrt{3} \left (x - \frac{1}{2}\right )^{2}} \]

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*s
qrt(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/XCAS [A]  time = 0.262192, size = 43, normalized size = 1.16 \[ \frac{2}{3} \,{\left (\sqrt{2 \, x - 1} + 3\right )}^{\frac{3}{2}} + 4 \, \sqrt{3} - 6 \, \sqrt{\sqrt{2 \, x - 1} + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(sqrt(2*x - 1) + 3),x, algorithm="giac")

[Out]

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