3.54 \(\int \frac{x^3 (1+x)^2}{\sqrt{1-x^2}} \, dx\)

Optimal. Leaf size=81 \[ -\frac{3}{5} \sqrt{1-x^2} x^2-\frac{3}{20} (5 x+8) \sqrt{1-x^2}-\frac{1}{5} \sqrt{1-x^2} x^4-\frac{1}{2} \sqrt{1-x^2} x^3+\frac{3}{4} \sin ^{-1}(x) \]

[Out]

(-3*x^2*Sqrt[1 - x^2])/5 - (x^3*Sqrt[1 - x^2])/2 - (x^4*Sqrt[1 - x^2])/5 - (3*(8
+ 5*x)*Sqrt[1 - x^2])/20 + (3*ArcSin[x])/4

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Rubi [A]  time = 0.215799, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{3}{5} \sqrt{1-x^2} x^2-\frac{3}{20} (5 x+8) \sqrt{1-x^2}-\frac{1}{5} \sqrt{1-x^2} x^4-\frac{1}{2} \sqrt{1-x^2} x^3+\frac{3}{4} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(x^3*(1 + x)^2)/Sqrt[1 - x^2],x]

[Out]

(-3*x^2*Sqrt[1 - x^2])/5 - (x^3*Sqrt[1 - x^2])/2 - (x^4*Sqrt[1 - x^2])/5 - (3*(8
+ 5*x)*Sqrt[1 - x^2])/20 + (3*ArcSin[x])/4

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Rubi in Sympy [A]  time = 15.0738, size = 61, normalized size = 0.75 \[ - \frac{x^{3} \sqrt{- x^{2} + 1}}{2} - \frac{3 x \sqrt{- x^{2} + 1}}{4} - \frac{\left (- x^{2} + 1\right )^{\frac{5}{2}}}{5} + \left (- x^{2} + 1\right )^{\frac{3}{2}} - 2 \sqrt{- x^{2} + 1} + \frac{3 \operatorname{asin}{\left (x \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x**3*sqrt(-x**2 + 1)/2 - 3*x*sqrt(-x**2 + 1)/4 - (-x**2 + 1)**(5/2)/5 + (-x**2
+ 1)**(3/2) - 2*sqrt(-x**2 + 1) + 3*asin(x)/4

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Mathematica [A]  time = 0.0483942, size = 77, normalized size = 0.95 \[ \frac{4 x^6+10 x^5+8 x^4+5 x^3+12 x^2-15 x+30 \sqrt{x-1} \sqrt{x+1} \log \left (\sqrt{x-1}+\sqrt{x+1}\right )-24}{20 \sqrt{1-x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^3*(1 + x)^2)/Sqrt[1 - x^2],x]

[Out]

(-24 - 15*x + 12*x^2 + 5*x^3 + 8*x^4 + 10*x^5 + 4*x^6 + 30*Sqrt[-1 + x]*Sqrt[1 +
x]*Log[Sqrt[-1 + x] + Sqrt[1 + x]])/(20*Sqrt[1 - x^2])

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Maple [A]  time = 0.027, size = 71, normalized size = 0.9 \[ -{\frac{3\,{x}^{2}}{5}\sqrt{-{x}^{2}+1}}-{\frac{6}{5}\sqrt{-{x}^{2}+1}}-{\frac{{x}^{4}}{5}\sqrt{-{x}^{2}+1}}-{\frac{{x}^{3}}{2}\sqrt{-{x}^{2}+1}}-{\frac{3\,x}{4}\sqrt{-{x}^{2}+1}}+{\frac{3\,\arcsin \left ( x \right ) }{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(1+x)^2/(-x^2+1)^(1/2),x)

[Out]

-3/5*x^2*(-x^2+1)^(1/2)-6/5*(-x^2+1)^(1/2)-1/5*x^4*(-x^2+1)^(1/2)-1/2*x^3*(-x^2+
1)^(1/2)-3/4*x*(-x^2+1)^(1/2)+3/4*arcsin(x)

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Maxima [A]  time = 0.787851, size = 95, normalized size = 1.17 \[ -\frac{1}{5} \, \sqrt{-x^{2} + 1} x^{4} - \frac{1}{2} \, \sqrt{-x^{2} + 1} x^{3} - \frac{3}{5} \, \sqrt{-x^{2} + 1} x^{2} - \frac{3}{4} \, \sqrt{-x^{2} + 1} x - \frac{6}{5} \, \sqrt{-x^{2} + 1} + \frac{3}{4} \, \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^2*x^3/sqrt(-x^2 + 1),x, algorithm="maxima")

[Out]

-1/5*sqrt(-x^2 + 1)*x^4 - 1/2*sqrt(-x^2 + 1)*x^3 - 3/5*sqrt(-x^2 + 1)*x^2 - 3/4*
sqrt(-x^2 + 1)*x - 6/5*sqrt(-x^2 + 1) + 3/4*arcsin(x)

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Fricas [A]  time = 0.271925, size = 239, normalized size = 2.95 \[ -\frac{4 \, x^{10} + 10 \, x^{9} - 40 \, x^{8} - 115 \, x^{7} - 20 \, x^{6} + 85 \, x^{5} + 80 \, x^{4} + 260 \, x^{3} + 30 \,{\left (5 \, x^{4} - 20 \, x^{2} -{\left (x^{4} - 12 \, x^{2} + 16\right )} \sqrt{-x^{2} + 1} + 16\right )} \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) + 5 \,{\left (4 \, x^{8} + 10 \, x^{7} - 4 \, x^{6} - 25 \, x^{5} - 16 \, x^{4} - 28 \, x^{3} + 48 \, x\right )} \sqrt{-x^{2} + 1} - 240 \, x}{20 \,{\left (5 \, x^{4} - 20 \, x^{2} -{\left (x^{4} - 12 \, x^{2} + 16\right )} \sqrt{-x^{2} + 1} + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^2*x^3/sqrt(-x^2 + 1),x, algorithm="fricas")

[Out]

-1/20*(4*x^10 + 10*x^9 - 40*x^8 - 115*x^7 - 20*x^6 + 85*x^5 + 80*x^4 + 260*x^3 +
30*(5*x^4 - 20*x^2 - (x^4 - 12*x^2 + 16)*sqrt(-x^2 + 1) + 16)*arctan((sqrt(-x^2
+ 1) - 1)/x) + 5*(4*x^8 + 10*x^7 - 4*x^6 - 25*x^5 - 16*x^4 - 28*x^3 + 48*x)*sqr
t(-x^2 + 1) - 240*x)/(5*x^4 - 20*x^2 - (x^4 - 12*x^2 + 16)*sqrt(-x^2 + 1) + 16)

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Sympy [A]  time = 3.48297, size = 73, normalized size = 0.9 \[ - \frac{x^{4} \sqrt{- x^{2} + 1}}{5} - \frac{x^{3} \sqrt{- x^{2} + 1}}{2} - \frac{3 x^{2} \sqrt{- x^{2} + 1}}{5} - \frac{3 x \sqrt{- x^{2} + 1}}{4} - \frac{6 \sqrt{- x^{2} + 1}}{5} + \frac{3 \operatorname{asin}{\left (x \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x**4*sqrt(-x**2 + 1)/5 - x**3*sqrt(-x**2 + 1)/2 - 3*x**2*sqrt(-x**2 + 1)/5 - 3*
x*sqrt(-x**2 + 1)/4 - 6*sqrt(-x**2 + 1)/5 + 3*asin(x)/4

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GIAC/XCAS [A]  time = 0.274286, size = 46, normalized size = 0.57 \[ -\frac{1}{20} \,{\left ({\left (2 \,{\left ({\left (2 \, x + 5\right )} x + 6\right )} x + 15\right )} x + 24\right )} \sqrt{-x^{2} + 1} + \frac{3}{4} \, \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^2*x^3/sqrt(-x^2 + 1),x, algorithm="giac")

[Out]

-1/20*((2*((2*x + 5)*x + 6)*x + 15)*x + 24)*sqrt(-x^2 + 1) + 3/4*arcsin(x)