\(\int \frac {\sqrt {-x+x^2} \sqrt {x^2-x \sqrt {-x+x^2}}}{x^3} \, dx\) [1796]

Optimal result

Integrand size = 37, antiderivative size = 121 \[ \int \frac {\sqrt {-x+x^2} \sqrt {x^2-x \sqrt {-x+x^2}}}{x^3} \, dx=-\frac {4 \sqrt {-x+x^2} \sqrt {x \left (x-\sqrt {-x+x^2}\right )}}{3 x^2}+\sqrt {x \left (x-\sqrt {-x+x^2}\right )} \left (\frac {4}{3 x}-\frac {2 \sqrt {2} \sqrt {x+\sqrt {-x+x^2}} \text {arctanh}\left (\sqrt {2} \sqrt {x+\sqrt {-x+x^2}}\right )}{x}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 2.47 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt {-x+x^2} \sqrt {x^2-x \sqrt {-x+x^2}}}{x^3} \, dx=-\frac {2 \sqrt {x \left (x-\sqrt {(-1+x) x}\right )} \left (2 x-2 \left (1+\sqrt {(-1+x) x}\right )+3 \sqrt {2} \sqrt {(-1+x) x} \sqrt {x+\sqrt {(-1+x) x}} \text {arctanh}\left (\sqrt {2} \sqrt {x+\sqrt {(-1+x) x}}\right )\right )}{3 x \sqrt {(-1+x) x}} \] Input:

Integrate[(Sqrt[-x + x^2]*Sqrt[x^2 - x*Sqrt[-x + x^2]])/x^3,x]
 

Output:

(-2*Sqrt[x*(x - Sqrt[(-1 + x)*x])]*(2*x - 2*(1 + Sqrt[(-1 + x)*x]) + 3*Sqr 
t[2]*Sqrt[(-1 + x)*x]*Sqrt[x + Sqrt[(-1 + x)*x]]*ArcTanh[Sqrt[2]*Sqrt[x + 
Sqrt[(-1 + x)*x]]]))/(3*x*Sqrt[(-1 + x)*x])
 

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.

\(\displaystyle \int \frac {\sqrt {x^2-x} \sqrt {x^2-x \sqrt {x^2-x}}}{x^3} \, dx\)

\(\Big \downarrow \) 2467

\(\displaystyle \frac {\sqrt {x^2-x} \int \frac {\sqrt {x-1} \sqrt {x^2-x \sqrt {x^2-x}}}{x^{5/2}}dx}{\sqrt {x-1} \sqrt {x}}\)

\(\Big \downarrow \) 2035

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

\(\Big \downarrow \) 7299

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

Input:

Int[(Sqrt[-x + x^2]*Sqrt[x^2 - x*Sqrt[-x + x^2]])/x^3,x]
 

Output:

$Aborted
 

Maple [F]

\[\int \frac {\sqrt {x^{2}-x}\, \sqrt {x^{2}-x \sqrt {x^{2}-x}}}{x^{3}}d x\]

Input:

int((x^2-x)^(1/2)*(x^2-x*(x^2-x)^(1/2))^(1/2)/x^3,x)
 

Output:

int((x^2-x)^(1/2)*(x^2-x*(x^2-x)^(1/2))^(1/2)/x^3,x)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {-x+x^2} \sqrt {x^2-x \sqrt {-x+x^2}}}{x^3} \, dx=\frac {3 \, \sqrt {2} x^{2} \log \left (-\frac {4 \, x^{2} - 2 \, \sqrt {x^{2} - \sqrt {x^{2} - x} x} {\left (\sqrt {2} x - \sqrt {2} \sqrt {x^{2} - x}\right )} - 4 \, \sqrt {x^{2} - x} x - x}{x}\right ) + 4 \, \sqrt {x^{2} - \sqrt {x^{2} - x} x} {\left (x - \sqrt {x^{2} - x}\right )}}{3 \, x^{2}} \] Input:

integrate((x^2-x)^(1/2)*(x^2-x*(x^2-x)^(1/2))^(1/2)/x^3,x, algorithm="fric 
as")
 

Output:

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

Sympy [F]

\[ \int \frac {\sqrt {-x+x^2} \sqrt {x^2-x \sqrt {-x+x^2}}}{x^3} \, dx=\int \frac {\sqrt {x \left (x - 1\right )} \sqrt {x \left (x - \sqrt {x^{2} - x}\right )}}{x^{3}}\, dx \] Input:

integrate((x**2-x)**(1/2)*(x**2-x*(x**2-x)**(1/2))**(1/2)/x**3,x)
 

Output:

Integral(sqrt(x*(x - 1))*sqrt(x*(x - sqrt(x**2 - x)))/x**3, x)
 

Maxima [F]

\[ \int \frac {\sqrt {-x+x^2} \sqrt {x^2-x \sqrt {-x+x^2}}}{x^3} \, dx=\int { \frac {\sqrt {x^{2} - \sqrt {x^{2} - x} x} \sqrt {x^{2} - x}}{x^{3}} \,d x } \] Input:

integrate((x^2-x)^(1/2)*(x^2-x*(x^2-x)^(1/2))^(1/2)/x^3,x, algorithm="maxi 
ma")
 

Output:

integrate(sqrt(x^2 - sqrt(x^2 - x)*x)*sqrt(x^2 - x)/x^3, x)
 

Giac [F]

\[ \int \frac {\sqrt {-x+x^2} \sqrt {x^2-x \sqrt {-x+x^2}}}{x^3} \, dx=\int { \frac {\sqrt {x^{2} - \sqrt {x^{2} - x} x} \sqrt {x^{2} - x}}{x^{3}} \,d x } \] Input:

integrate((x^2-x)^(1/2)*(x^2-x*(x^2-x)^(1/2))^(1/2)/x^3,x, algorithm="giac 
")
 

Output:

integrate(sqrt(x^2 - sqrt(x^2 - x)*x)*sqrt(x^2 - x)/x^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {-x+x^2} \sqrt {x^2-x \sqrt {-x+x^2}}}{x^3} \, dx=\int \frac {\sqrt {x^2-x}\,\sqrt {x^2-x\,\sqrt {x^2-x}}}{x^3} \,d x \] Input:

int(((x^2 - x)^(1/2)*(x^2 - x*(x^2 - x)^(1/2))^(1/2))/x^3,x)
 

Output:

int(((x^2 - x)^(1/2)*(x^2 - x*(x^2 - x)^(1/2))^(1/2))/x^3, x)
 

Reduce [F]

\[ \int \frac {\sqrt {-x+x^2} \sqrt {x^2-x \sqrt {-x+x^2}}}{x^3} \, dx=\int \frac {\sqrt {x^{2}-x}\, \sqrt {x^{2}-x \sqrt {x^{2}-x}}}{x^{3}}d x \] Input:

int((x^2-x)^(1/2)*(x^2-x*(x^2-x)^(1/2))^(1/2)/x^3,x)
 

Output:

int((x^2-x)^(1/2)*(x^2-x*(x^2-x)^(1/2))^(1/2)/x^3,x)