3.559 \(\int \sqrt{2-\sqrt{4+\sqrt{-9+5 x}}} \, dx\)

Optimal. Leaf size=82 \[ \frac{8}{45} \left (2-\sqrt{\sqrt{5 x-9}+4}\right )^{9/2}-\frac{48}{35} \left (2-\sqrt{\sqrt{5 x-9}+4}\right )^{7/2}+\frac{64}{25} \left (2-\sqrt{\sqrt{5 x-9}+4}\right )^{5/2} \]

[Out]

(64*(2 - Sqrt[4 + Sqrt[-9 + 5*x]])^(5/2))/25 - (48*(2 - Sqrt[4 + Sqrt[-9 + 5*x]]
)^(7/2))/35 + (8*(2 - Sqrt[4 + Sqrt[-9 + 5*x]])^(9/2))/45

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Rubi [A]  time = 0.167852, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ \frac{8}{45} \left (2-\sqrt{\sqrt{5 x-9}+4}\right )^{9/2}-\frac{48}{35} \left (2-\sqrt{\sqrt{5 x-9}+4}\right )^{7/2}+\frac{64}{25} \left (2-\sqrt{\sqrt{5 x-9}+4}\right )^{5/2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[2 - Sqrt[4 + Sqrt[-9 + 5*x]]],x]

[Out]

(64*(2 - Sqrt[4 + Sqrt[-9 + 5*x]])^(5/2))/25 - (48*(2 - Sqrt[4 + Sqrt[-9 + 5*x]]
)^(7/2))/35 + (8*(2 - Sqrt[4 + Sqrt[-9 + 5*x]])^(9/2))/45

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Rubi in Sympy [A]  time = 6.56881, size = 65, normalized size = 0.79 \[ \frac{8 \left (- \sqrt{\sqrt{5 x - 9} + 4} + 2\right )^{\frac{9}{2}}}{45} - \frac{48 \left (- \sqrt{\sqrt{5 x - 9} + 4} + 2\right )^{\frac{7}{2}}}{35} + \frac{64 \left (- \sqrt{\sqrt{5 x - 9} + 4} + 2\right )^{\frac{5}{2}}}{25} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2-(4+(-9+5*x)**(1/2))**(1/2))**(1/2),x)

[Out]

8*(-sqrt(sqrt(5*x - 9) + 4) + 2)**(9/2)/45 - 48*(-sqrt(sqrt(5*x - 9) + 4) + 2)**
(7/2)/35 + 64*(-sqrt(sqrt(5*x - 9) + 4) + 2)**(5/2)/25

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Mathematica [A]  time = 0.0452686, size = 86, normalized size = 1.05 \[ -\frac{8 \sqrt{2-\sqrt{\sqrt{5 x-9}+4}} \left (-175 x-4 \sqrt{5 x-9}+10 \sqrt{5 x-9} \sqrt{\sqrt{5 x-9}+4}-64 \sqrt{\sqrt{5 x-9}+4}+443\right )}{1575} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[2 - Sqrt[4 + Sqrt[-9 + 5*x]]],x]

[Out]

(-8*Sqrt[2 - Sqrt[4 + Sqrt[-9 + 5*x]]]*(443 - 175*x - 4*Sqrt[-9 + 5*x] - 64*Sqrt
[4 + Sqrt[-9 + 5*x]] + 10*Sqrt[-9 + 5*x]*Sqrt[4 + Sqrt[-9 + 5*x]]))/1575

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Maple [A]  time = 0.015, size = 59, normalized size = 0.7 \[{\frac{64}{25} \left ( 2-\sqrt{4+\sqrt{-9+5\,x}} \right ) ^{{\frac{5}{2}}}}-{\frac{48}{35} \left ( 2-\sqrt{4+\sqrt{-9+5\,x}} \right ) ^{{\frac{7}{2}}}}+{\frac{8}{45} \left ( 2-\sqrt{4+\sqrt{-9+5\,x}} \right ) ^{{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2-(4+(-9+5*x)^(1/2))^(1/2))^(1/2),x)

[Out]

64/25*(2-(4+(-9+5*x)^(1/2))^(1/2))^(5/2)-48/35*(2-(4+(-9+5*x)^(1/2))^(1/2))^(7/2
)+8/45*(2-(4+(-9+5*x)^(1/2))^(1/2))^(9/2)

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Maxima [A]  time = 0.723031, size = 78, normalized size = 0.95 \[ \frac{8}{45} \,{\left (-\sqrt{\sqrt{5 \, x - 9} + 4} + 2\right )}^{\frac{9}{2}} - \frac{48}{35} \,{\left (-\sqrt{\sqrt{5 \, x - 9} + 4} + 2\right )}^{\frac{7}{2}} + \frac{64}{25} \,{\left (-\sqrt{\sqrt{5 \, x - 9} + 4} + 2\right )}^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-sqrt(sqrt(5*x - 9) + 4) + 2),x, algorithm="maxima")

[Out]

8/45*(-sqrt(sqrt(5*x - 9) + 4) + 2)^(9/2) - 48/35*(-sqrt(sqrt(5*x - 9) + 4) + 2)
^(7/2) + 64/25*(-sqrt(sqrt(5*x - 9) + 4) + 2)^(5/2)

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Fricas [A]  time = 0.267317, size = 77, normalized size = 0.94 \[ -\frac{8}{1575} \,{\left (2 \,{\left (5 \, \sqrt{5 \, x - 9} - 32\right )} \sqrt{\sqrt{5 \, x - 9} + 4} - 175 \, x - 4 \, \sqrt{5 \, x - 9} + 443\right )} \sqrt{-\sqrt{\sqrt{5 \, x - 9} + 4} + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-sqrt(sqrt(5*x - 9) + 4) + 2),x, algorithm="fricas")

[Out]

-8/1575*(2*(5*sqrt(5*x - 9) - 32)*sqrt(sqrt(5*x - 9) + 4) - 175*x - 4*sqrt(5*x -
9) + 443)*sqrt(-sqrt(sqrt(5*x - 9) + 4) + 2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{- \sqrt{\sqrt{5 x - 9} + 4} + 2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2-(4+(-9+5*x)**(1/2))**(1/2))**(1/2),x)

[Out]

Integral(sqrt(-sqrt(sqrt(5*x - 9) + 4) + 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(sqrt(-sqrt(sqrt(5*x - 9) + 4) + 2),x, algorithm="giac")

[Out]

Exception raised: TypeError