3.558 \(\int \sqrt{2+\sqrt{4+\sqrt{x}}} \, dx\)

Optimal. Leaf size=64 \[ \frac{8}{9} \left (\sqrt{\sqrt{x}+4}+2\right )^{9/2}-\frac{48}{7} \left (\sqrt{\sqrt{x}+4}+2\right )^{7/2}+\frac{64}{5} \left (\sqrt{\sqrt{x}+4}+2\right )^{5/2} \]

[Out]

(64*(2 + Sqrt[4 + Sqrt[x]])^(5/2))/5 - (48*(2 + Sqrt[4 + Sqrt[x]])^(7/2))/7 + (8
*(2 + Sqrt[4 + Sqrt[x]])^(9/2))/9

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Rubi [A]  time = 0.10365, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \frac{8}{9} \left (\sqrt{\sqrt{x}+4}+2\right )^{9/2}-\frac{48}{7} \left (\sqrt{\sqrt{x}+4}+2\right )^{7/2}+\frac{64}{5} \left (\sqrt{\sqrt{x}+4}+2\right )^{5/2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[2 + Sqrt[4 + Sqrt[x]]],x]

[Out]

(64*(2 + Sqrt[4 + Sqrt[x]])^(5/2))/5 - (48*(2 + Sqrt[4 + Sqrt[x]])^(7/2))/7 + (8
*(2 + Sqrt[4 + Sqrt[x]])^(9/2))/9

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Rubi in Sympy [A]  time = 4.79705, size = 54, normalized size = 0.84 \[ \frac{8 \left (\sqrt{\sqrt{x} + 4} + 2\right )^{\frac{9}{2}}}{9} - \frac{48 \left (\sqrt{\sqrt{x} + 4} + 2\right )^{\frac{7}{2}}}{7} + \frac{64 \left (\sqrt{\sqrt{x} + 4} + 2\right )^{\frac{5}{2}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+(4+x**(1/2))**(1/2))**(1/2),x)

[Out]

8*(sqrt(sqrt(x) + 4) + 2)**(9/2)/9 - 48*(sqrt(sqrt(x) + 4) + 2)**(7/2)/7 + 64*(s
qrt(sqrt(x) + 4) + 2)**(5/2)/5

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Mathematica [A]  time = 0.0395384, size = 62, normalized size = 0.97 \[ \frac{8}{315} \sqrt{\sqrt{\sqrt{x}+4}+2} \left (-64 \left (\sqrt{\sqrt{x}+4}+2\right )+35 x+2 \left (5 \sqrt{\sqrt{x}+4}+2\right ) \sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[2 + Sqrt[4 + Sqrt[x]]],x]

[Out]

(8*Sqrt[2 + Sqrt[4 + Sqrt[x]]]*(-64*(2 + Sqrt[4 + Sqrt[x]]) + 2*(2 + 5*Sqrt[4 +
Sqrt[x]])*Sqrt[x] + 35*x))/315

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Maple [A]  time = 0.014, size = 41, normalized size = 0.6 \[{\frac{64}{5} \left ( 2+\sqrt{4+\sqrt{x}} \right ) ^{{\frac{5}{2}}}}-{\frac{48}{7} \left ( 2+\sqrt{4+\sqrt{x}} \right ) ^{{\frac{7}{2}}}}+{\frac{8}{9} \left ( 2+\sqrt{4+\sqrt{x}} \right ) ^{{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+(4+x^(1/2))^(1/2))^(1/2),x)

[Out]

64/5*(2+(4+x^(1/2))^(1/2))^(5/2)-48/7*(2+(4+x^(1/2))^(1/2))^(7/2)+8/9*(2+(4+x^(1
/2))^(1/2))^(9/2)

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Maxima [A]  time = 0.733047, size = 54, normalized size = 0.84 \[ \frac{8}{9} \,{\left (\sqrt{\sqrt{x} + 4} + 2\right )}^{\frac{9}{2}} - \frac{48}{7} \,{\left (\sqrt{\sqrt{x} + 4} + 2\right )}^{\frac{7}{2}} + \frac{64}{5} \,{\left (\sqrt{\sqrt{x} + 4} + 2\right )}^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(sqrt(x) + 4) + 2),x, algorithm="maxima")

[Out]

8/9*(sqrt(sqrt(x) + 4) + 2)^(9/2) - 48/7*(sqrt(sqrt(x) + 4) + 2)^(7/2) + 64/5*(s
qrt(sqrt(x) + 4) + 2)^(5/2)

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Fricas [A]  time = 0.265884, size = 53, normalized size = 0.83 \[ \frac{8}{315} \,{\left (2 \,{\left (5 \, \sqrt{x} - 32\right )} \sqrt{\sqrt{x} + 4} + 35 \, x + 4 \, \sqrt{x} - 128\right )} \sqrt{\sqrt{\sqrt{x} + 4} + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(sqrt(x) + 4) + 2),x, algorithm="fricas")

[Out]

8/315*(2*(5*sqrt(x) - 32)*sqrt(sqrt(x) + 4) + 35*x + 4*sqrt(x) - 128)*sqrt(sqrt(
sqrt(x) + 4) + 2)

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Sympy [A]  time = 8.12604, size = 216, normalized size = 3.38 \[ - \frac{2 \sqrt{2} \sqrt{x} \sqrt{\sqrt{x} + 4} \sqrt{\sqrt{\sqrt{x} + 4} + 2} \Gamma \left (- \frac{1}{4}\right ) \Gamma \left (\frac{1}{4}\right )}{63 \pi } - \frac{4 \sqrt{2} \sqrt{x} \sqrt{\sqrt{\sqrt{x} + 4} + 2} \Gamma \left (- \frac{1}{4}\right ) \Gamma \left (\frac{1}{4}\right )}{315 \pi } - \frac{\sqrt{2} x \sqrt{\sqrt{\sqrt{x} + 4} + 2} \Gamma \left (- \frac{1}{4}\right ) \Gamma \left (\frac{1}{4}\right )}{9 \pi } + \frac{64 \sqrt{2} \sqrt{\sqrt{x} + 4} \sqrt{\sqrt{\sqrt{x} + 4} + 2} \Gamma \left (- \frac{1}{4}\right ) \Gamma \left (\frac{1}{4}\right )}{315 \pi } + \frac{128 \sqrt{2} \sqrt{\sqrt{\sqrt{x} + 4} + 2} \Gamma \left (- \frac{1}{4}\right ) \Gamma \left (\frac{1}{4}\right )}{315 \pi } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2+(4+x**(1/2))**(1/2))**(1/2),x)

[Out]

-2*sqrt(2)*sqrt(x)*sqrt(sqrt(x) + 4)*sqrt(sqrt(sqrt(x) + 4) + 2)*gamma(-1/4)*gam
ma(1/4)/(63*pi) - 4*sqrt(2)*sqrt(x)*sqrt(sqrt(sqrt(x) + 4) + 2)*gamma(-1/4)*gamm
a(1/4)/(315*pi) - sqrt(2)*x*sqrt(sqrt(sqrt(x) + 4) + 2)*gamma(-1/4)*gamma(1/4)/(
9*pi) + 64*sqrt(2)*sqrt(sqrt(x) + 4)*sqrt(sqrt(sqrt(x) + 4) + 2)*gamma(-1/4)*gam
ma(1/4)/(315*pi) + 128*sqrt(2)*sqrt(sqrt(sqrt(x) + 4) + 2)*gamma(-1/4)*gamma(1/4
)/(315*pi)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(sqrt(x) + 4) + 2),x, algorithm="giac")

[Out]

Exception raised: TypeError