3.858 \(\int \frac{\sqrt{-x+\sqrt{x} \sqrt{1+x}}}{\sqrt{1+x}} \, dx\)

Optimal. Leaf size=66 \[ \frac{1}{2} \left (\sqrt{x}+3 \sqrt{x+1}\right ) \sqrt{\sqrt{x} \sqrt{x+1}-x}-\frac{3 \sin ^{-1}\left (\sqrt{x}-\sqrt{x+1}\right )}{2 \sqrt{2}} \]

[Out]

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

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Rubi [F]  time = 0.229311, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0. \[ \text{Int}\left (\frac{\sqrt{-x+\sqrt{x} \sqrt{1+x}}}{\sqrt{1+x}},x\right ) \]

Verification is Not applicable to the result.

[In]  Int[Sqrt[-x + Sqrt[x]*Sqrt[1 + x]]/Sqrt[1 + x],x]

[Out]

2*Defer[Subst][Defer[Int][Sqrt[1 - x^2 + x*Sqrt[-1 + x^2]], x], x, Sqrt[1 + x]]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*Integral(sqrt(-x**2 + x*sqrt(x**2 - 1) + 1), (x, sqrt(x + 1)))

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Mathematica [B]  time = 0.703212, size = 180, normalized size = 2.73 \[ -\frac{(x+1) \left (2 x-2 \sqrt{x+1} \sqrt{x}+1\right )^2 \left (2 \sqrt{\sqrt{x} \sqrt{x+1}-x} \left (-2 x+2 \sqrt{x+1} \sqrt{x}-3\right )+3 \sqrt{-4 x+4 \sqrt{x+1} \sqrt{x}-2} \log \left (2 \sqrt{\sqrt{x} \sqrt{x+1}-x}+\sqrt{-4 x+4 \sqrt{x+1} \sqrt{x}-2}\right )\right )}{4 \left (\sqrt{x+1}-\sqrt{x}\right )^3 \left (x-\sqrt{x+1} \sqrt{x}+1\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[-x + Sqrt[x]*Sqrt[1 + x]]/Sqrt[1 + x],x]

[Out]

-((1 + x)*(1 + 2*x - 2*Sqrt[x]*Sqrt[1 + x])^2*(2*Sqrt[-x + Sqrt[x]*Sqrt[1 + x]]*
(-3 - 2*x + 2*Sqrt[x]*Sqrt[1 + x]) + 3*Sqrt[-2 - 4*x + 4*Sqrt[x]*Sqrt[1 + x]]*Lo
g[2*Sqrt[-x + Sqrt[x]*Sqrt[1 + x]] + Sqrt[-2 - 4*x + 4*Sqrt[x]*Sqrt[1 + x]]]))/(
4*(-Sqrt[x] + Sqrt[1 + x])^3*(1 + x - Sqrt[x]*Sqrt[1 + x])^2)

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Maple [F]  time = 0.031, size = 0, normalized size = 0. \[ \int{1\sqrt{-x+\sqrt{x}\sqrt{1+x}}{\frac{1}{\sqrt{1+x}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\sqrt{x + 1} \sqrt{x} - x}}{\sqrt{x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(sqrt(x + 1)*sqrt(x) - x)/sqrt(x + 1), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(sqrt(x)*sqrt(x + 1) - x)/sqrt(x + 1), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\sqrt{x + 1} \sqrt{x} - x}}{\sqrt{x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(sqrt(x + 1)*sqrt(x) - x)/sqrt(x + 1), x)