3.557 \(\int \frac{1}{\sqrt{2+\sqrt{1+\sqrt{x}}}} \, dx\)

Optimal. Leaf size=83 \[ \frac{8}{7} \left (\sqrt{\sqrt{x}+1}+2\right )^{7/2}-\frac{48}{5} \left (\sqrt{\sqrt{x}+1}+2\right )^{5/2}+\frac{88}{3} \left (\sqrt{\sqrt{x}+1}+2\right )^{3/2}-48 \sqrt{\sqrt{\sqrt{x}+1}+2} \]

[Out]

-48*Sqrt[2 + Sqrt[1 + Sqrt[x]]] + (88*(2 + Sqrt[1 + Sqrt[x]])^(3/2))/3 - (48*(2
+ Sqrt[1 + Sqrt[x]])^(5/2))/5 + (8*(2 + Sqrt[1 + Sqrt[x]])^(7/2))/7

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Rubi [A]  time = 0.114372, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{8}{7} \left (\sqrt{\sqrt{x}+1}+2\right )^{7/2}-\frac{48}{5} \left (\sqrt{\sqrt{x}+1}+2\right )^{5/2}+\frac{88}{3} \left (\sqrt{\sqrt{x}+1}+2\right )^{3/2}-48 \sqrt{\sqrt{\sqrt{x}+1}+2} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[2 + Sqrt[1 + Sqrt[x]]],x]

[Out]

-48*Sqrt[2 + Sqrt[1 + Sqrt[x]]] + (88*(2 + Sqrt[1 + Sqrt[x]])^(3/2))/3 - (48*(2
+ Sqrt[1 + Sqrt[x]])^(5/2))/5 + (8*(2 + Sqrt[1 + Sqrt[x]])^(7/2))/7

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Rubi in Sympy [A]  time = 4.68887, size = 71, normalized size = 0.86 \[ \frac{8 \left (\sqrt{\sqrt{x} + 1} + 2\right )^{\frac{7}{2}}}{7} - \frac{48 \left (\sqrt{\sqrt{x} + 1} + 2\right )^{\frac{5}{2}}}{5} + \frac{88 \left (\sqrt{\sqrt{x} + 1} + 2\right )^{\frac{3}{2}}}{3} - 48 \sqrt{\sqrt{\sqrt{x} + 1} + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8*(sqrt(sqrt(x) + 1) + 2)**(7/2)/7 - 48*(sqrt(sqrt(x) + 1) + 2)**(5/2)/5 + 88*(s
qrt(sqrt(x) + 1) + 2)**(3/2)/3 - 48*sqrt(sqrt(sqrt(x) + 1) + 2)

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Mathematica [A]  time = 0.0425017, size = 58, normalized size = 0.7 \[ \frac{8}{105} \sqrt{\sqrt{\sqrt{x}+1}+2} \left (3 \sqrt{x} \left (5 \sqrt{\sqrt{x}+1}-12\right )+76 \sqrt{\sqrt{x}+1}-280\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[2 + Sqrt[1 + Sqrt[x]]],x]

[Out]

(8*Sqrt[2 + Sqrt[1 + Sqrt[x]]]*(-280 + 76*Sqrt[1 + Sqrt[x]] + 3*(-12 + 5*Sqrt[1
+ Sqrt[x]])*Sqrt[x]))/105

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Maple [A]  time = 0.013, size = 54, normalized size = 0.7 \[{\frac{88}{3} \left ( 2+\sqrt{1+\sqrt{x}} \right ) ^{{\frac{3}{2}}}}-{\frac{48}{5} \left ( 2+\sqrt{1+\sqrt{x}} \right ) ^{{\frac{5}{2}}}}+{\frac{8}{7} \left ( 2+\sqrt{1+\sqrt{x}} \right ) ^{{\frac{7}{2}}}}-48\,\sqrt{2+\sqrt{1+\sqrt{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

88/3*(2+(1+x^(1/2))^(1/2))^(3/2)-48/5*(2+(1+x^(1/2))^(1/2))^(5/2)+8/7*(2+(1+x^(1
/2))^(1/2))^(7/2)-48*(2+(1+x^(1/2))^(1/2))^(1/2)

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Maxima [A]  time = 0.741382, size = 72, normalized size = 0.87 \[ \frac{8}{7} \,{\left (\sqrt{\sqrt{x} + 1} + 2\right )}^{\frac{7}{2}} - \frac{48}{5} \,{\left (\sqrt{\sqrt{x} + 1} + 2\right )}^{\frac{5}{2}} + \frac{88}{3} \,{\left (\sqrt{\sqrt{x} + 1} + 2\right )}^{\frac{3}{2}} - 48 \, \sqrt{\sqrt{\sqrt{x} + 1} + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(sqrt(sqrt(x) + 1) + 2),x, algorithm="maxima")

[Out]

8/7*(sqrt(sqrt(x) + 1) + 2)^(7/2) - 48/5*(sqrt(sqrt(x) + 1) + 2)^(5/2) + 88/3*(s
qrt(sqrt(x) + 1) + 2)^(3/2) - 48*sqrt(sqrt(sqrt(x) + 1) + 2)

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Fricas [A]  time = 0.269128, size = 47, normalized size = 0.57 \[ \frac{8}{105} \,{\left ({\left (15 \, \sqrt{x} + 76\right )} \sqrt{\sqrt{x} + 1} - 36 \, \sqrt{x} - 280\right )} \sqrt{\sqrt{\sqrt{x} + 1} + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8/105*((15*sqrt(x) + 76)*sqrt(sqrt(x) + 1) - 36*sqrt(x) - 280)*sqrt(sqrt(sqrt(x)
+ 1) + 2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{\sqrt{\sqrt{x} + 1} + 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/sqrt(sqrt(sqrt(x) + 1) + 2), x)

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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(1/sqrt(sqrt(sqrt(x) + 1) + 2),x, algorithm="giac")

[Out]

Exception raised: TypeError