Integrand size = 37, antiderivative size = 121 \[ \int \frac {\sqrt {-x+x^2} \sqrt {x^2-x \sqrt {-x+x^2}}}{x^3} \, dx=-\frac {4 \sqrt {-x+x^2} \sqrt {x \left (x-\sqrt {-x+x^2}\right )}}{3 x^2}+\sqrt {x \left (x-\sqrt {-x+x^2}\right )} \left (\frac {4}{3 x}-\frac {2 \sqrt {2} \sqrt {x+\sqrt {-x+x^2}} \text {arctanh}\left (\sqrt {2} \sqrt {x+\sqrt {-x+x^2}}\right )}{x}\right ) \]
-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)
Time = 2.52 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt {-x+x^2} \sqrt {x^2-x \sqrt {-x+x^2}}}{x^3} \, dx=-\frac {2 \sqrt {x \left (x-\sqrt {(-1+x) x}\right )} \left (2 x-2 \left (1+\sqrt {(-1+x) x}\right )+3 \sqrt {2} \sqrt {(-1+x) x} \sqrt {x+\sqrt {(-1+x) x}} \text {arctanh}\left (\sqrt {2} \sqrt {x+\sqrt {(-1+x) x}}\right )\right )}{3 x \sqrt {(-1+x) x}} \]
(-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])
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 |
\(\displaystyle \frac {2 \sqrt {x^2-x} \int \frac {\sqrt {x-1} \sqrt {x^2-x \sqrt {x^2-x}}}{x^2}d\sqrt {x}}{\sqrt {x-1} \sqrt {x}}\) |
3.18.96.3.1 Defintions of rubi rules used
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
\[\int \frac {\sqrt {x^{2}-x}\, \sqrt {x^{2}-x \sqrt {x^{2}-x}}}{x^{3}}d x\]
Time = 0.28 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {-x+x^2} \sqrt {x^2-x \sqrt {-x+x^2}}}{x^3} \, dx=\frac {3 \, \sqrt {2} x^{2} \log \left (-\frac {4 \, x^{2} - 2 \, \sqrt {x^{2} - \sqrt {x^{2} - x} x} {\left (\sqrt {2} x - \sqrt {2} \sqrt {x^{2} - x}\right )} - 4 \, \sqrt {x^{2} - x} x - x}{x}\right ) + 4 \, \sqrt {x^{2} - \sqrt {x^{2} - x} x} {\left (x - \sqrt {x^{2} - x}\right )}}{3 \, x^{2}} \]
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
\[ \int \frac {\sqrt {-x+x^2} \sqrt {x^2-x \sqrt {-x+x^2}}}{x^3} \, dx=\int \frac {\sqrt {x \left (x - 1\right )} \sqrt {x \left (x - \sqrt {x^{2} - x}\right )}}{x^{3}}\, dx \]
\[ \int \frac {\sqrt {-x+x^2} \sqrt {x^2-x \sqrt {-x+x^2}}}{x^3} \, dx=\int { \frac {\sqrt {x^{2} - \sqrt {x^{2} - x} x} \sqrt {x^{2} - x}}{x^{3}} \,d x } \]
\[ \int \frac {\sqrt {-x+x^2} \sqrt {x^2-x \sqrt {-x+x^2}}}{x^3} \, dx=\int { \frac {\sqrt {x^{2} - \sqrt {x^{2} - x} x} \sqrt {x^{2} - x}}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {-x+x^2} \sqrt {x^2-x \sqrt {-x+x^2}}}{x^3} \, dx=\int \frac {\sqrt {x^2-x}\,\sqrt {x^2-x\,\sqrt {x^2-x}}}{x^3} \,d x \]