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

Optimal. Leaf size=73 \[ \frac {4 (x-3) \sqrt {-x \left (\sqrt {x^2-x}-x\right )}}{15 x^2}-\frac {4 \sqrt {x^2-x} \sqrt {-x \left (\sqrt {x^2-x}-x\right )}}{15 x^2} \]

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Rubi [F]  time = 0.17, 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^2-x \sqrt {-x+x^2}}}{x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[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*} \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\\ \end {align*}

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Mathematica [A]  time = 0.21, size = 58, normalized size = 0.79 \begin {gather*} \frac {4 x \left (4 x^2-\left (4 \sqrt {(x-1) x}+9\right ) x+7 \sqrt {(x-1) x}+3\right )}{15 \left (x \left (x-\sqrt {(x-1) x}\right )\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*x*(3 + 4*x^2 + 7*Sqrt[(-1 + x)*x] - x*(9 + 4*Sqrt[(-1 + x)*x])))/(15*(x*(x - Sqrt[(-1 + x)*x]))^(3/2))

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IntegrateAlgebraic [A]  time = 3.88, size = 73, normalized size = 1.00 \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.46, size = 37, normalized size = 0.51 \begin {gather*} \frac {4 \, \sqrt {x^{2} - \sqrt {x^{2} - x} x} {\left (x - \sqrt {x^{2} - x} - 3\right )}}{15 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {x^{2}-x \sqrt {x^{2}-x}}}{x^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2-x*(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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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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}}}{x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\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*(x**2-x)**(1/2))**(1/2)/x**3,x)

[Out]

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

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