\(\int \frac {\sqrt {-x+\sqrt {x} \sqrt {1+x}}}{\sqrt {1+x}} \, dx\) [362]

Optimal result

Integrand size = 29, antiderivative size = 88 \[ \int \frac {\sqrt {-x+\sqrt {x} \sqrt {1+x}}}{\sqrt {1+x}} \, dx=\frac {1}{2} \sqrt {x} \sqrt {-x+\sqrt {x} \sqrt {1+x}}+\frac {3}{2} \sqrt {1+x} \sqrt {-x+\sqrt {x} \sqrt {1+x}}-\frac {3 \arcsin \left (\sqrt {x}-\sqrt {1+x}\right )}{2 \sqrt {2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.07 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.03 \[ \int \frac {\sqrt {-x+\sqrt {x} \sqrt {1+x}}}{\sqrt {1+x}} \, dx=\frac {1}{4} \left (2 \left (\sqrt {x}+3 \sqrt {1+x}\right ) \sqrt {-x+\sqrt {x} \sqrt {1+x}}-3 \sqrt {2} \arctan \left (\frac {\sqrt {-2 x+2 \sqrt {x} \sqrt {1+x}}}{-\sqrt {x}+\sqrt {1+x}}\right )\right ) \] Input:

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

Output:

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

Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 7267

\(\displaystyle 2 \int \sqrt {\sqrt {x} \sqrt {x+1}-x}d\sqrt {x+1}\)

\(\Big \downarrow \) 7299

\(\displaystyle 2 \int \sqrt {\sqrt {x} \sqrt {x+1}-x}d\sqrt {x+1}\)

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 7267

Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si 
mp[lst[[2]]*lst[[4]]   Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x 
] /;  !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
 

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

integrate((-x+x**(1/2)*(1+x)**(1/2))**(1/2)/(1+x)**(1/2),x)
 

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {-x+\sqrt {x} \sqrt {1+x}}}{\sqrt {1+x}} \, dx=\int \frac {x^{\frac {1}{4}} \sqrt {x +1}\, \sqrt {\sqrt {x +1}-\sqrt {x}}}{x +1}d x \] Input:

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

Output:

int((x**(1/4)*sqrt(x + 1)*sqrt(sqrt(x + 1) - sqrt(x)))/(x + 1),x)