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

Optimal. Leaf size=121 \[ \sqrt {x \left (x-\sqrt {x^2-x}\right )} \left (\frac {4}{3 x}-\frac {2 \sqrt {2} \sqrt {\sqrt {x^2-x}+x} \tanh ^{-1}\left (\sqrt {2} \sqrt {\sqrt {x^2-x}+x}\right )}{x}\right )-\frac {4 \sqrt {x^2-x} \sqrt {x \left (x-\sqrt {x^2-x}\right )}}{3 x^2} \]

________________________________________________________________________________________

Rubi [F]  time = 2.01, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\sqrt {-x+x^2} \sqrt {x^2-x \sqrt {-x+x^2}}}{x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[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*} \int \frac {\sqrt {-x+x^2} \sqrt {x^2-x \sqrt {-x+x^2}}}{x^3} \, dx &=\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 ) \operatorname {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]  time = 0.88, size = 175, normalized size = 1.45 \begin {gather*} \frac {2 \sqrt {(x-1) x} \left (-16 x^2+16 \left (\sqrt {(x-1) x}+1\right ) x-8 \sqrt {(x-1) x}+3 \sqrt {4 x-4 \sqrt {(x-1) x}-2} \left (x-\sqrt {(x-1) x}\right )^{3/2} \log \left (\sqrt {4 x-4 \sqrt {(x-1) x}-2}+2 \sqrt {x-\sqrt {(x-1) x}}\right )-2\right )}{3 \sqrt {x \left (x-\sqrt {(x-1) x}\right )} \left (\sqrt {(x-1) x}-x\right ) \left (-x+\sqrt {(x-1) x}+1\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[(-1 + x)*x]*(-2 - 16*x^2 - 8*Sqrt[(-1 + x)*x] + 16*x*(1 + Sqrt[(-1 + x)*x]) + 3*Sqrt[-2 + 4*x - 4*Sqrt
[(-1 + x)*x]]*(x - Sqrt[(-1 + x)*x])^(3/2)*Log[Sqrt[-2 + 4*x - 4*Sqrt[(-1 + x)*x]] + 2*Sqrt[x - Sqrt[(-1 + x)*
x]]]))/(3*Sqrt[x*(x - Sqrt[(-1 + x)*x])]*(-x + Sqrt[(-1 + x)*x])*(1 - x + Sqrt[(-1 + x)*x]))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 4.22, size = 121, normalized size = 1.00 \begin {gather*} -\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}} \tanh ^{-1}\left (\sqrt {2} \sqrt {x+\sqrt {-x+x^2}}\right )}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*Sqrt[-x + x^2]*Sqrt[x*(x - Sqrt[-x + x^2])])/(3*x^2) + Sqrt[x*(x - Sqrt[-x + x^2])]*(4/(3*x) - (2*Sqrt[2]*
Sqrt[x + Sqrt[-x + x^2]]*ArcTanh[Sqrt[2]*Sqrt[x + Sqrt[-x + x^2]]])/x)

________________________________________________________________________________________

fricas [A]  time = 0.49, size = 114, normalized size = 0.94 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{2} - \sqrt {x^{2} - x} x} \sqrt {x^{2} - x}}{x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {x^{2}-x}\, \sqrt {x^{2}-x \sqrt {x^{2}-x}}}{x^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{2} - \sqrt {x^{2} - x} x} \sqrt {x^{2} - x}}{x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {x^2-x}\,\sqrt {x^2-x\,\sqrt {x^2-x}}}{x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x \left (x - 1\right )} \sqrt {x \left (x - \sqrt {x^{2} - x}\right )}}{x^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________