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

Optimal. Leaf size=63 \[ -\frac{2}{3} \sqrt{1-x^2} x^2-\frac{1}{24} (21 x+32) \sqrt{1-x^2}-\frac{1}{4} \sqrt{1-x^2} x^3+\frac{7}{8} \sin ^{-1}(x) \]

[Out]

(-2*x^2*Sqrt[1 - x^2])/3 - (x^3*Sqrt[1 - x^2])/4 - ((32 + 21*x)*Sqrt[1 - x^2])/2
4 + (7*ArcSin[x])/8

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Rubi [A]  time = 0.161405, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{2}{3} \sqrt{1-x^2} x^2-\frac{1}{24} (21 x+32) \sqrt{1-x^2}-\frac{1}{4} \sqrt{1-x^2} x^3+\frac{7}{8} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

(-2*x^2*Sqrt[1 - x^2])/3 - (x^3*Sqrt[1 - x^2])/4 - ((32 + 21*x)*Sqrt[1 - x^2])/2
4 + (7*ArcSin[x])/8

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Rubi in Sympy [A]  time = 14.2894, size = 54, normalized size = 0.86 \[ - \frac{x^{3} \sqrt{- x^{2} + 1}}{4} - \frac{7 x \sqrt{- x^{2} + 1}}{8} + \frac{2 \left (- x^{2} + 1\right )^{\frac{3}{2}}}{3} - 2 \sqrt{- x^{2} + 1} + \frac{7 \operatorname{asin}{\left (x \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x**3*sqrt(-x**2 + 1)/4 - 7*x*sqrt(-x**2 + 1)/8 + 2*(-x**2 + 1)**(3/2)/3 - 2*sqr
t(-x**2 + 1) + 7*asin(x)/8

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Mathematica [A]  time = 0.0428156, size = 72, normalized size = 1.14 \[ \frac{6 x^5+16 x^4+15 x^3+16 x^2-21 x+42 \sqrt{x-1} \sqrt{x+1} \log \left (\sqrt{x-1}+\sqrt{x+1}\right )-32}{24 \sqrt{1-x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-32 - 21*x + 16*x^2 + 15*x^3 + 16*x^4 + 6*x^5 + 42*Sqrt[-1 + x]*Sqrt[1 + x]*Log
[Sqrt[-1 + x] + Sqrt[1 + x]])/(24*Sqrt[1 - x^2])

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Maple [A]  time = 0.008, size = 57, normalized size = 0.9 \[ -{\frac{7\,x}{8}\sqrt{-{x}^{2}+1}}+{\frac{7\,\arcsin \left ( x \right ) }{8}}-{\frac{{x}^{3}}{4}\sqrt{-{x}^{2}+1}}-{\frac{2\,{x}^{2}}{3}\sqrt{-{x}^{2}+1}}-{\frac{4}{3}\sqrt{-{x}^{2}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7/8*x*(-x^2+1)^(1/2)+7/8*arcsin(x)-1/4*x^3*(-x^2+1)^(1/2)-2/3*x^2*(-x^2+1)^(1/2
)-4/3*(-x^2+1)^(1/2)

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Maxima [A]  time = 0.782387, size = 76, normalized size = 1.21 \[ -\frac{1}{4} \, \sqrt{-x^{2} + 1} x^{3} - \frac{2}{3} \, \sqrt{-x^{2} + 1} x^{2} - \frac{7}{8} \, \sqrt{-x^{2} + 1} x - \frac{4}{3} \, \sqrt{-x^{2} + 1} + \frac{7}{8} \, \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*sqrt(-x^2 + 1)*x^3 - 2/3*sqrt(-x^2 + 1)*x^2 - 7/8*sqrt(-x^2 + 1)*x - 4/3*sq
rt(-x^2 + 1) + 7/8*arcsin(x)

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Fricas [A]  time = 0.269787, size = 193, normalized size = 3.06 \[ \frac{24 \, x^{7} + 64 \, x^{6} + 12 \, x^{5} - 96 \, x^{4} - 204 \, x^{3} - 42 \,{\left (x^{4} - 8 \, x^{2} + 4 \,{\left (x^{2} - 2\right )} \sqrt{-x^{2} + 1} + 8\right )} \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) -{\left (6 \, x^{7} + 16 \, x^{6} - 27 \, x^{5} - 96 \, x^{4} - 120 \, x^{3} + 168 \, x\right )} \sqrt{-x^{2} + 1} + 168 \, x}{24 \,{\left (x^{4} - 8 \, x^{2} + 4 \,{\left (x^{2} - 2\right )} \sqrt{-x^{2} + 1} + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*(24*x^7 + 64*x^6 + 12*x^5 - 96*x^4 - 204*x^3 - 42*(x^4 - 8*x^2 + 4*(x^2 - 2
)*sqrt(-x^2 + 1) + 8)*arctan((sqrt(-x^2 + 1) - 1)/x) - (6*x^7 + 16*x^6 - 27*x^5
- 96*x^4 - 120*x^3 + 168*x)*sqrt(-x^2 + 1) + 168*x)/(x^4 - 8*x^2 + 4*(x^2 - 2)*s
qrt(-x^2 + 1) + 8)

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Sympy [A]  time = 1.98461, size = 60, normalized size = 0.95 \[ - \frac{x^{3} \sqrt{- x^{2} + 1}}{4} - \frac{2 x^{2} \sqrt{- x^{2} + 1}}{3} - \frac{7 x \sqrt{- x^{2} + 1}}{8} - \frac{4 \sqrt{- x^{2} + 1}}{3} + \frac{7 \operatorname{asin}{\left (x \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x**3*sqrt(-x**2 + 1)/4 - 2*x**2*sqrt(-x**2 + 1)/3 - 7*x*sqrt(-x**2 + 1)/8 - 4*s
qrt(-x**2 + 1)/3 + 7*asin(x)/8

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GIAC/XCAS [A]  time = 0.283878, size = 41, normalized size = 0.65 \[ -\frac{1}{24} \,{\left ({\left (2 \,{\left (3 \, x + 8\right )} x + 21\right )} x + 32\right )} \sqrt{-x^{2} + 1} + \frac{7}{8} \, \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/24*((2*(3*x + 8)*x + 21)*x + 32)*sqrt(-x^2 + 1) + 7/8*arcsin(x)