3.792 \(\int \sqrt{x^2+x^3-x^4} \, dx\)

Optimal. Leaf size=107 \[ -\frac{\sqrt{-x^4+x^3+x^2} (1-2 x)}{8 x}-\frac{\left (-x^2+x+1\right ) \sqrt{-x^4+x^3+x^2}}{3 x}-\frac{5 \sqrt{-x^4+x^3+x^2} \sin ^{-1}\left (\frac{1-2 x}{\sqrt{5}}\right )}{16 x \sqrt{-x^2+x+1}} \]

[Out]

-((1 - 2*x)*Sqrt[x^2 + x^3 - x^4])/(8*x) - ((1 + x - x^2)*Sqrt[x^2 + x^3 - x^4])
/(3*x) - (5*Sqrt[x^2 + x^3 - x^4]*ArcSin[(1 - 2*x)/Sqrt[5]])/(16*x*Sqrt[1 + x -
x^2])

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Rubi [A]  time = 0.0637217, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312 \[ -\frac{\sqrt{-x^4+x^3+x^2} (1-2 x)}{8 x}-\frac{\left (-x^2+x+1\right ) \sqrt{-x^4+x^3+x^2}}{3 x}-\frac{5 \sqrt{-x^4+x^3+x^2} \sin ^{-1}\left (\frac{1-2 x}{\sqrt{5}}\right )}{16 x \sqrt{-x^2+x+1}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x^2 + x^3 - x^4],x]

[Out]

-((1 - 2*x)*Sqrt[x^2 + x^3 - x^4])/(8*x) - ((1 + x - x^2)*Sqrt[x^2 + x^3 - x^4])
/(3*x) - (5*Sqrt[x^2 + x^3 - x^4]*ArcSin[(1 - 2*x)/Sqrt[5]])/(16*x*Sqrt[1 + x -
x^2])

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Rubi in Sympy [A]  time = 5.45391, size = 94, normalized size = 0.88 \[ - \frac{\left (- 2 x + 1\right ) \sqrt{- x^{4} + x^{3} + x^{2}}}{8 x} - \frac{\left (- x^{2} + x + 1\right ) \sqrt{- x^{4} + x^{3} + x^{2}}}{3 x} - \frac{5 \sqrt{- x^{4} + x^{3} + x^{2}} \operatorname{atan}{\left (\frac{- 2 x + 1}{2 \sqrt{- x^{2} + x + 1}} \right )}}{16 x \sqrt{- x^{2} + x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-x**4+x**3+x**2)**(1/2),x)

[Out]

-(-2*x + 1)*sqrt(-x**4 + x**3 + x**2)/(8*x) - (-x**2 + x + 1)*sqrt(-x**4 + x**3
+ x**2)/(3*x) - 5*sqrt(-x**4 + x**3 + x**2)*atan((-2*x + 1)/(2*sqrt(-x**2 + x +
1)))/(16*x*sqrt(-x**2 + x + 1))

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Mathematica [A]  time = 0.0438319, size = 82, normalized size = 0.77 \[ \frac{\sqrt{-x^4+x^3+x^2} \left (2 \sqrt{x^2-x-1} \left (8 x^2-2 x-11\right )-15 \log \left (-2 \sqrt{x^2-x-1}-2 x+1\right )\right )}{48 x \sqrt{x^2-x-1}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x^2 + x^3 - x^4],x]

[Out]

(Sqrt[x^2 + x^3 - x^4]*(2*Sqrt[-1 - x + x^2]*(-11 - 2*x + 8*x^2) - 15*Log[1 - 2*
x - 2*Sqrt[-1 - x + x^2]]))/(48*x*Sqrt[-1 - x + x^2])

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Maple [A]  time = 0.009, size = 81, normalized size = 0.8 \[{\frac{1}{48\,x}\sqrt{-{x}^{4}+{x}^{3}+{x}^{2}} \left ( -16\, \left ( -{x}^{2}+x+1 \right ) ^{3/2}+12\,x\sqrt{-{x}^{2}+x+1}+15\,\arcsin \left ( 1/5\, \left ( 2\,x-1 \right ) \sqrt{5} \right ) -6\,\sqrt{-{x}^{2}+x+1} \right ){\frac{1}{\sqrt{-{x}^{2}+x+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^4+x^3+x^2)^(1/2),x)

[Out]

1/48*(-x^4+x^3+x^2)^(1/2)*(-16*(-x^2+x+1)^(3/2)+12*x*(-x^2+x+1)^(1/2)+15*arcsin(
1/5*(2*x-1)*5^(1/2))-6*(-x^2+x+1)^(1/2))/x/(-x^2+x+1)^(1/2)

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Maxima [A]  time = 0.794795, size = 69, normalized size = 0.64 \[ -\frac{1}{3} \,{\left (-x^{2} + x + 1\right )}^{\frac{3}{2}} + \frac{1}{4} \, \sqrt{-x^{2} + x + 1} x - \frac{1}{8} \, \sqrt{-x^{2} + x + 1} - \frac{5}{16} \, \arcsin \left (-\frac{1}{5} \, \sqrt{5}{\left (2 \, x - 1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^4 + x^3 + x^2),x, algorithm="maxima")

[Out]

-1/3*(-x^2 + x + 1)^(3/2) + 1/4*sqrt(-x^2 + x + 1)*x - 1/8*sqrt(-x^2 + x + 1) -
5/16*arcsin(-1/5*sqrt(5)*(2*x - 1))

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Fricas [A]  time = 0.281575, size = 84, normalized size = 0.79 \[ -\frac{15 \, x \arctan \left (-\frac{x - \sqrt{-x^{4} + x^{3} + x^{2}}}{x^{2}}\right ) - \sqrt{-x^{4} + x^{3} + x^{2}}{\left (8 \, x^{2} - 2 \, x - 11\right )} + 11 \, x}{24 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^4 + x^3 + x^2),x, algorithm="fricas")

[Out]

-1/24*(15*x*arctan(-(x - sqrt(-x^4 + x^3 + x^2))/x^2) - sqrt(-x^4 + x^3 + x^2)*(
8*x^2 - 2*x - 11) + 11*x)/x

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{- x^{4} + x^{3} + x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x**4+x**3+x**2)**(1/2),x)

[Out]

Integral(sqrt(-x**4 + x**3 + x**2), x)

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GIAC/XCAS [A]  time = 0.268193, size = 81, normalized size = 0.76 \[ \frac{1}{48} \,{\left (15 \, \arcsin \left (\frac{1}{5} \, \sqrt{5}\right ) + 22\right )}{\rm sign}\left (x\right ) + \frac{5}{16} \, \arcsin \left (\frac{1}{5} \, \sqrt{5}{\left (2 \, x - 1\right )}\right ){\rm sign}\left (x\right ) + \frac{1}{24} \,{\left (2 \,{\left (4 \, x{\rm sign}\left (x\right ) -{\rm sign}\left (x\right )\right )} x - 11 \,{\rm sign}\left (x\right )\right )} \sqrt{-x^{2} + x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^4 + x^3 + x^2),x, algorithm="giac")

[Out]

1/48*(15*arcsin(1/5*sqrt(5)) + 22)*sign(x) + 5/16*arcsin(1/5*sqrt(5)*(2*x - 1))*
sign(x) + 1/24*(2*(4*x*sign(x) - sign(x))*x - 11*sign(x))*sqrt(-x^2 + x + 1)