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

Optimal. Leaf size=131 \[ \frac {\sqrt {x^2-x} \sqrt {-x \left (\sqrt {x^2-x}-x\right )} (8 x-9)}{12 x}+\sqrt {x \left (x-\sqrt {x^2-x}\right )} \left (\frac {3 \sqrt {\sqrt {x^2-x}+x} \tanh ^{-1}\left (\sqrt {2} \sqrt {\sqrt {x^2-x}+x}\right )}{4 \sqrt {2} x}+\frac {1}{12} (8 x-19)\right ) \]

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Rubi [F]  time = 1.89, 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}}} \, 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]

[Out]

(2*Sqrt[-x + x^2]*Defer[Subst][Defer[Int][(x^2*Sqrt[-1 + x^2])/Sqrt[x^4 - x^2*Sqrt[-x^2 + 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}}} \, dx &=\frac {\sqrt {-x+x^2} \int \frac {\sqrt {-1+x} \sqrt {x}}{\sqrt {x^2-x \sqrt {-x+x^2}}} \, dx}{\sqrt {-1+x} \sqrt {x}}\\ &=\frac {\left (2 \sqrt {-x+x^2}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {-1+x^2}}{\sqrt {x^4-x^2 \sqrt {-x^2+x^4}}} \, dx,x,\sqrt {x}\right )}{\sqrt {-1+x} \sqrt {x}}\\ \end {align*}

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Mathematica [C]  time = 0.39, size = 160, normalized size = 1.22 \begin {gather*} \frac {\sqrt {(x-1) x} \left (\sqrt {(x-1) x}-x\right ) \left (9 \left (-2 x+2 \sqrt {(x-1) x}+1\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};1+\frac {1}{2 \left (\sqrt {(x-1) x}-x\right )}\right )-24 x^2+24 \sqrt {(x-1) x} x+22 x-10 \sqrt {(x-1) x}\right )}{12 \sqrt {x \left (x-\sqrt {(x-1) x}\right )} \left (4 x^2-\left (4 \sqrt {(x-1) x}+5\right ) x+3 \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]

[Out]

(Sqrt[(-1 + x)*x]*(-x + Sqrt[(-1 + x)*x])*(22*x - 24*x^2 - 10*Sqrt[(-1 + x)*x] + 24*x*Sqrt[(-1 + x)*x] + 9*(1
- 2*x + 2*Sqrt[(-1 + x)*x])*Hypergeometric2F1[-1/2, 1, 1/2, 1 + 1/(2*(-x + Sqrt[(-1 + x)*x]))]))/(12*Sqrt[x*(x
- Sqrt[(-1 + x)*x])]*(1 + 4*x^2 + 3*Sqrt[(-1 + x)*x] - x*(5 + 4*Sqrt[(-1 + x)*x])))

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IntegrateAlgebraic [A]  time = 4.29, size = 131, normalized size = 1.00 \begin {gather*} \frac {(-9+8 x) \sqrt {-x+x^2} \sqrt {-x \left (-x+\sqrt {-x+x^2}\right )}}{12 x}+\sqrt {x \left (x-\sqrt {-x+x^2}\right )} \left (\frac {1}{12} (-19+8 x)+\frac {3 \sqrt {x+\sqrt {-x+x^2}} \tanh ^{-1}\left (\sqrt {2} \sqrt {x+\sqrt {-x+x^2}}\right )}{4 \sqrt {2} x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-9 + 8*x)*Sqrt[-x + x^2]*Sqrt[-(x*(-x + Sqrt[-x + x^2]))])/(12*x) + Sqrt[x*(x - Sqrt[-x + x^2])]*((-19 + 8*x
)/12 + (3*Sqrt[x + Sqrt[-x + x^2]]*ArcTanh[Sqrt[2]*Sqrt[x + Sqrt[-x + x^2]]])/(4*Sqrt[2]*x))

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fricas [A]  time = 0.47, size = 123, normalized size = 0.94 \begin {gather*} \frac {9 \, \sqrt {2} x \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 \, {\left (8 \, x^{2} + \sqrt {x^{2} - x} {\left (8 \, x - 9\right )} - 19 \, x\right )} \sqrt {x^{2} - \sqrt {x^{2} - x} x}}{48 \, 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, algorithm="fricas")

[Out]

1/48*(9*sqrt(2)*x*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*(8*x^2 + sqrt(x^2 - x)*(8*x - 9) - 19*x)*sqrt(x^2 - sqrt(x^2 - x)*x))/x

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

[Out]

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

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

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)

[Out]

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

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

[Out]

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

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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}}} \,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)

[Out]

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

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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 )}}\, 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)

[Out]

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

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