\(\int \frac {\sqrt {x^2-x \sqrt {-x+x^2}}}{x^3} \, dx\) [966]

Optimal result

Integrand size = 26, antiderivative size = 73 \[ \int \frac {\sqrt {x^2-x \sqrt {-x+x^2}}}{x^3} \, dx=\frac {4 (-3+x) \sqrt {-x \left (-x+\sqrt {-x+x^2}\right )}}{15 x^2}-\frac {4 \sqrt {-x+x^2} \sqrt {-x \left (-x+\sqrt {-x+x^2}\right )}}{15 x^2} \]

[Out]

4/15*(-3+x)*(-x*(-x+(x^2-x)^(1/2)))^(1/2)/x^2-4/15*(x^2-x)^(1/2)*(-x*(-x+(x^2-x)^(1/2)))^(1/2)/x^2

Rubi [F]

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

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt {x^2-x \sqrt {-x+x^2}}}{x^3} \, dx \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 1.21 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.25 \[ \int \frac {\sqrt {x^2-x \sqrt {-x+x^2}}}{x^3} \, dx=-\frac {4 \sqrt {x \left (x-\sqrt {(-1+x) x}\right )} \sqrt {x+\sqrt {(-1+x) x}} \left (-2 x^2-3 \sqrt {(-1+x) x}+2 x \left (2+\sqrt {(-1+x) x}\right )\right )}{15 x^{5/2} \sqrt {x-\sqrt {(-1+x) x}}} \]

[In]

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

[Out]

(-4*Sqrt[x*(x - Sqrt[(-1 + x)*x])]*Sqrt[x + Sqrt[(-1 + x)*x]]*(-2*x^2 - 3*Sqrt[(-1 + x)*x] + 2*x*(2 + Sqrt[(-1
+ x)*x])))/(15*x^(5/2)*Sqrt[x - Sqrt[(-1 + x)*x]])

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.51 \[ \int \frac {\sqrt {x^2-x \sqrt {-x+x^2}}}{x^3} \, dx=\frac {4 \, \sqrt {x^{2} - \sqrt {x^{2} - x} x} {\left (x - \sqrt {x^{2} - x} - 3\right )}}{15 \, x^{2}} \]

[In]

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

[Out]

4/15*sqrt(x^2 - sqrt(x^2 - x)*x)*(x - sqrt(x^2 - x) - 3)/x^2

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

integrate((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)/x^3, x)

Giac [F]

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

[In]

integrate((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)/x^3, x)

Mupad [F(-1)]

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

[In]

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

[Out]

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