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

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 = 0.138587, 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., Rules used = {} \[ \int \frac{\sqrt{-x+\sqrt{x} \sqrt{1+x}}}{\sqrt{1+x}} \, dx \]

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]]

Rubi steps

\begin{align*} \int \frac{\sqrt{-x+\sqrt{x} \sqrt{1+x}}}{\sqrt{1+x}} \, dx &=2 \operatorname{Subst}\left (\int \sqrt{1-x^2+x \sqrt{-1+x^2}} \, dx,x,\sqrt{1+x}\right )\\ \end{align*}

Mathematica [B]  time = 0.522494, 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]]*Log[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 = 0.019, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{-x+\sqrt{x}\sqrt{1+x}}{\frac{1}{\sqrt{1+x}}}}\, dx \end{align*}

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. \begin{align*} \int \frac{\sqrt{\sqrt{x + 1} \sqrt{x} - x}}{\sqrt{x + 1}}\,{d x} \end{align*}

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, 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. \begin{align*} \text{Timed out} \end{align*}

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, algorithm="fricas")

[Out]

Timed out

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

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

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, algorithm="giac")

[Out]

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