3.759 \(\int \sqrt{x-x^2} F(x) \, dx\)

Optimal. Leaf size=17 \[ \text{Int}\left (\sqrt{x-x^2} F(x),x\right ) \]

[Out]

CannotIntegrate[Sqrt[x - x^2]*F[x], x]

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Rubi [A]  time = 0.0571509, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0. \[ \text{Int}\left (\sqrt{x-x^2} F(x),x\right ) \]

Verification is Not applicable to the result.

[In]  Int[Sqrt[x - x^2]*F[x],x]

[Out]

Defer[Int][Sqrt[x - x^2]*F[x], x]

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Rubi in Sympy [A]  time = 0., size = 0, normalized size = 0. \[ \frac{2 \sqrt{- x^{2} + x} \int ^{\sqrt{x}} x^{2} \sqrt{- x^{2} + 1} F{\left (x^{2} \right )}\, dx}{\sqrt{x} \sqrt{- x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F(x)*(-x**2+x)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.100756, size = 0, normalized size = 0. \[ \int \sqrt{x-x^2} F(x) \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[Sqrt[x - x^2]*F[x],x]

[Out]

Integrate[Sqrt[x - x^2]*F[x], x]

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Maple [A]  time = 0.026, size = 0, normalized size = 0. \[ \int F \left ( x \right ) \sqrt{-{x}^{2}+x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F(x)*(-x^2+x)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{-x^{2} + x} F\left (x\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^2 + x)*F(x),x, algorithm="maxima")

[Out]

integrate(sqrt(-x^2 + x)*F(x), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^2 + x)*F(x),x, algorithm="fricas")

[Out]

Exception raised: TypeError

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Sympy [A]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{- x \left (x - 1\right )} F{\left (x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F(x)*(-x**2+x)**(1/2),x)

[Out]

Integral(sqrt(-x*(x - 1))*F(x), x)

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GIAC/XCAS [A]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{-x^{2} + x} F\left (x\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^2 + x)*F(x),x, algorithm="giac")

[Out]

integrate(sqrt(-x^2 + x)*F(x), x)