\(\int \frac {\sqrt {-x+\sqrt {x} \sqrt {1+x}}}{\sqrt {1+x}} \, dx\) [996]

Optimal result

Integrand size = 29, antiderivative size = 66 \[ \int \frac {\sqrt {-x+\sqrt {x} \sqrt {1+x}}}{\sqrt {1+x}} \, dx=\frac {1}{2} \left (\sqrt {x}+3 \sqrt {1+x}\right ) \sqrt {-x+\sqrt {x} \sqrt {1+x}}-\frac {3 \arcsin \left (\sqrt {x}-\sqrt {1+x}\right )}{2 \sqrt {2}} \]

[Out]

-3/4*arcsin(x^(1/2)-(1+x)^(1/2))*2^(1/2)+1/2*(x^(1/2)+3*(1+x)^(1/2))*(-x+x^(1/2)*(1+x)^(1/2))^(1/2)

Rubi [F]

\[ \int \frac {\sqrt {-x+\sqrt {x} \sqrt {1+x}}}{\sqrt {1+x}} \, dx=\int \frac {\sqrt {-x+\sqrt {x} \sqrt {1+x}}}{\sqrt {1+x}} \, dx \]

[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*} \text {integral}& = 2 \text {Subst}\left (\int \sqrt {1-x^2+x \sqrt {-1+x^2}} \, dx,x,\sqrt {1+x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.62 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.38 \[ \int \frac {\sqrt {-x+\sqrt {x} \sqrt {1+x}}}{\sqrt {1+x}} \, dx=\frac {1}{4} \left (2 \left (\sqrt {x}+3 \sqrt {1+x}\right ) \sqrt {-x+\sqrt {x} \sqrt {1+x}}-3 \sqrt {2} \arctan \left (\frac {\sqrt {-2 x+2 \sqrt {x} \sqrt {1+x}}}{-\sqrt {x}+\sqrt {1+x}}\right )\right ) \]

[In]

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

[Out]

(2*(Sqrt[x] + 3*Sqrt[1 + x])*Sqrt[-x + Sqrt[x]*Sqrt[1 + x]] - 3*Sqrt[2]*ArcTan[Sqrt[-2*x + 2*Sqrt[x]*Sqrt[1 +
x]]/(-Sqrt[x] + Sqrt[1 + x])])/4

Maple [F]

\[\int \frac {\sqrt {-x +\sqrt {x}\, \sqrt {x +1}}}{\sqrt {x +1}}d x\]

[In]

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

[Out]

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

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {-x+\sqrt {x} \sqrt {1+x}}}{\sqrt {1+x}} \, dx=\text {Timed out} \]

[In]

integrate((-x+x^(1/2)*(1+x)^(1/2))^(1/2)/(1+x)^(1/2),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int \frac {\sqrt {-x+\sqrt {x} \sqrt {1+x}}}{\sqrt {1+x}} \, dx=\int \frac {\sqrt {\sqrt {x} \sqrt {x + 1} - x}}{\sqrt {x + 1}}\, dx \]

[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)

Maxima [F]

\[ \int \frac {\sqrt {-x+\sqrt {x} \sqrt {1+x}}}{\sqrt {1+x}} \, dx=\int { \frac {\sqrt {\sqrt {x + 1} \sqrt {x} - x}}{\sqrt {x + 1}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {\sqrt {-x+\sqrt {x} \sqrt {1+x}}}{\sqrt {1+x}} \, dx=\int { \frac {\sqrt {\sqrt {x + 1} \sqrt {x} - x}}{\sqrt {x + 1}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {-x+\sqrt {x} \sqrt {1+x}}}{\sqrt {1+x}} \, dx=\int \frac {\sqrt {\sqrt {x}\,\sqrt {x+1}-x}}{\sqrt {x+1}} \,d x \]

[In]

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

[Out]

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