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

Optimal result

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

[Out]

-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)

Rubi [F]

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

[In]

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

[Out]

(2*Sqrt[-x + x^2]*Defer[Subst][Defer[Int][(Sqrt[-1 + x^2]*Sqrt[x^4 - x^2*Sqrt[-x^2 + x^4]])/x^4, x], x, Sqrt[x
]])/(Sqrt[-1 + x]*Sqrt[x])

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {-x+x^2} \int \frac {\sqrt {-1+x} \sqrt {x^2-x \sqrt {-x+x^2}}}{x^{5/2}} \, dx}{\sqrt {-1+x} \sqrt {x}} \\ & = \frac {\left (2 \sqrt {-x+x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {-1+x^2} \sqrt {x^4-x^2 \sqrt {-x^2+x^4}}}{x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {-1+x} \sqrt {x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.52 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt {-x+x^2} \sqrt {x^2-x \sqrt {-x+x^2}}}{x^3} \, dx=-\frac {2 \sqrt {x \left (x-\sqrt {(-1+x) x}\right )} \left (2 x-2 \left (1+\sqrt {(-1+x) x}\right )+3 \sqrt {2} \sqrt {(-1+x) x} \sqrt {x+\sqrt {(-1+x) x}} \text {arctanh}\left (\sqrt {2} \sqrt {x+\sqrt {(-1+x) x}}\right )\right )}{3 x \sqrt {(-1+x) x}} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {-x+x^2} \sqrt {x^2-x \sqrt {-x+x^2}}}{x^3} \, dx=\frac {3 \, \sqrt {2} x^{2} \log \left (-\frac {4 \, x^{2} - 2 \, \sqrt {x^{2} - \sqrt {x^{2} - x} x} {\left (\sqrt {2} x - \sqrt {2} \sqrt {x^{2} - x}\right )} - 4 \, \sqrt {x^{2} - x} x - x}{x}\right ) + 4 \, \sqrt {x^{2} - \sqrt {x^{2} - x} x} {\left (x - \sqrt {x^{2} - x}\right )}}{3 \, x^{2}} \]

[In]

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

[Out]

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]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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