3.100 \(\int \frac{x^2}{\sqrt{1-(1+x)^2}} \, dx\)

Optimal. Leaf size=44 \[ -\frac{1}{2} \sqrt{1-(x+1)^2} x+\frac{3}{2} \sqrt{1-(x+1)^2}+\frac{3}{2} \sin ^{-1}(x+1) \]

[Out]

(3*Sqrt[1 - (1 + x)^2])/2 - (x*Sqrt[1 - (1 + x)^2])/2 + (3*ArcSin[1 + x])/2

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Rubi [A]  time = 0.0657623, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ -\frac{1}{2} \sqrt{1-(x+1)^2} x+\frac{3}{2} \sqrt{1-(x+1)^2}+\frac{3}{2} \sin ^{-1}(x+1) \]

Antiderivative was successfully verified.

[In]  Int[x^2/Sqrt[1 - (1 + x)^2],x]

[Out]

(3*Sqrt[1 - (1 + x)^2])/2 - (x*Sqrt[1 - (1 + x)^2])/2 + (3*ArcSin[1 + x])/2

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Rubi in Sympy [A]  time = 8.27844, size = 34, normalized size = 0.77 \[ - \frac{x \sqrt{- \left (x + 1\right )^{2} + 1}}{2} + \frac{3 \sqrt{- \left (x + 1\right )^{2} + 1}}{2} + \frac{3 \operatorname{asin}{\left (x + 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x*sqrt(-(x + 1)**2 + 1)/2 + 3*sqrt(-(x + 1)**2 + 1)/2 + 3*asin(x + 1)/2

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Mathematica [A]  time = 0.0327615, size = 51, normalized size = 1.16 \[ \frac{x \left (x^2-x-6\right )+6 \sqrt{x} \sqrt{x+2} \sinh ^{-1}\left (\frac{\sqrt{x}}{\sqrt{2}}\right )}{2 \sqrt{-x (x+2)}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/Sqrt[1 - (1 + x)^2],x]

[Out]

(x*(-6 - x + x^2) + 6*Sqrt[x]*Sqrt[2 + x]*ArcSinh[Sqrt[x]/Sqrt[2]])/(2*Sqrt[-(x*
(2 + x))])

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Maple [A]  time = 0.006, size = 35, normalized size = 0.8 \[ -{\frac{x}{2}\sqrt{-{x}^{2}-2\,x}}+{\frac{3}{2}\sqrt{-{x}^{2}-2\,x}}+{\frac{3\,\arcsin \left ( 1+x \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(1-(1+x)^2)^(1/2),x)

[Out]

-1/2*x*(-x^2-2*x)^(1/2)+3/2*(-x^2-2*x)^(1/2)+3/2*arcsin(1+x)

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Maxima [A]  time = 0.890398, size = 49, normalized size = 1.11 \[ -\frac{1}{2} \, \sqrt{-x^{2} - 2 \, x} x + \frac{3}{2} \, \sqrt{-x^{2} - 2 \, x} - \frac{3}{2} \, \arcsin \left (-x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/sqrt(-(x + 1)^2 + 1),x, algorithm="maxima")

[Out]

-1/2*sqrt(-x^2 - 2*x)*x + 3/2*sqrt(-x^2 - 2*x) - 3/2*arcsin(-x - 1)

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Fricas [A]  time = 0.280235, size = 47, normalized size = 1.07 \[ -\frac{1}{2} \, \sqrt{-x^{2} - 2 \, x}{\left (x - 3\right )} - 3 \, \arctan \left (\frac{\sqrt{-x^{2} - 2 \, x}}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/sqrt(-(x + 1)^2 + 1),x, algorithm="fricas")

[Out]

-1/2*sqrt(-x^2 - 2*x)*(x - 3) - 3*arctan(sqrt(-x^2 - 2*x)/x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**2/sqrt(-x*(x + 2)), x)

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GIAC/XCAS [A]  time = 0.266137, size = 31, normalized size = 0.7 \[ -\frac{1}{2} \, \sqrt{-{\left (x + 1\right )}^{2} + 1}{\left (x - 3\right )} + \frac{3}{2} \, \arcsin \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/sqrt(-(x + 1)^2 + 1),x, algorithm="giac")

[Out]

-1/2*sqrt(-(x + 1)^2 + 1)*(x - 3) + 3/2*arcsin(x + 1)