3.606 \(\int x^3 (1+x)^3 (1+2 x) \sqrt{1-x^2-2 x^3-x^4} \, dx\)

Optimal. Leaf size=42 \[ -\frac{1}{15} \left (-x^4-2 x^3-x^2+1\right )^{3/2} \left (3 x^4+6 x^3+3 x^2+2\right ) \]

[Out]

-((1 - x^2 - 2*x^3 - x^4)^(3/2)*(2 + 3*x^2 + 6*x^3 + 3*x^4))/15

_______________________________________________________________________________________

Rubi [A]  time = 0.393367, antiderivative size = 59, normalized size of antiderivative = 1.4, number of steps used = 5, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{1}{5} x^2 \left (-x^4-2 x^3-x^2+1\right )^{3/2} (x+1)^2-\frac{2}{15} \left (-x^4-2 x^3-x^2+1\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[x^3*(1 + x)^3*(1 + 2*x)*Sqrt[1 - x^2 - 2*x^3 - x^4],x]

[Out]

(-2*(1 - x^2 - 2*x^3 - x^4)^(3/2))/15 - (x^2*(1 + x)^2*(1 - x^2 - 2*x^3 - x^4)^(
3/2))/5

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 23.4367, size = 60, normalized size = 1.43 \[ - \frac{\left (- 4 \left (x + \frac{1}{2}\right )^{2} + 1\right )^{2} \left (- 16 \left (x + \frac{1}{2}\right )^{4} + 8 \left (x + \frac{1}{2}\right )^{2} + 15\right )^{\frac{3}{2}}}{5120} - \frac{\left (- 16 \left (x + \frac{1}{2}\right )^{4} + 8 \left (x + \frac{1}{2}\right )^{2} + 15\right )^{\frac{3}{2}}}{480} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-4*(x + 1/2)**2 + 1)**2*(-16*(x + 1/2)**4 + 8*(x + 1/2)**2 + 15)**(3/2)/5120 -
(-16*(x + 1/2)**4 + 8*(x + 1/2)**2 + 15)**(3/2)/480

_______________________________________________________________________________________

Mathematica [A]  time = 0.0827473, size = 62, normalized size = 1.48 \[ \frac{1}{15} \sqrt{-x^4-2 x^3-x^2+1} \left (3 x^8+12 x^7+18 x^6+12 x^5+2 x^4-2 x^3-x^2-2\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*(1 + x)^3*(1 + 2*x)*Sqrt[1 - x^2 - 2*x^3 - x^4],x]

[Out]

(Sqrt[1 - x^2 - 2*x^3 - x^4]*(-2 - x^2 - 2*x^3 + 2*x^4 + 12*x^5 + 18*x^6 + 12*x^
7 + 3*x^8))/15

_______________________________________________________________________________________

Maple [A]  time = 0.011, size = 51, normalized size = 1.2 \[{\frac{ \left ({x}^{2}+x+1 \right ) \left ({x}^{2}+x-1 \right ) \left ( 3\,{x}^{4}+6\,{x}^{3}+3\,{x}^{2}+2 \right ) }{15}\sqrt{-{x}^{4}-2\,{x}^{3}-{x}^{2}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(x^2+x+1)*(x^2+x-1)*(3*x^4+6*x^3+3*x^2+2)*(-x^4-2*x^3-x^2+1)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.884715, size = 80, normalized size = 1.9 \[ \frac{1}{15} \,{\left (3 \, x^{8} + 12 \, x^{7} + 18 \, x^{6} + 12 \, x^{5} + 2 \, x^{4} - 2 \, x^{3} - x^{2} - 2\right )} \sqrt{x^{2} + x + 1} \sqrt{-x^{2} - x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^4 - 2*x^3 - x^2 + 1)*(2*x + 1)*(x + 1)^3*x^3,x, algorithm="maxima")

[Out]

1/15*(3*x^8 + 12*x^7 + 18*x^6 + 12*x^5 + 2*x^4 - 2*x^3 - x^2 - 2)*sqrt(x^2 + x +
1)*sqrt(-x^2 - x + 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.260599, size = 78, normalized size = 1.86 \[ \frac{1}{15} \,{\left (3 \, x^{8} + 12 \, x^{7} + 18 \, x^{6} + 12 \, x^{5} + 2 \, x^{4} - 2 \, x^{3} - x^{2} - 2\right )} \sqrt{-x^{4} - 2 \, x^{3} - x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^4 - 2*x^3 - x^2 + 1)*(2*x + 1)*(x + 1)^3*x^3,x, algorithm="fricas")

[Out]

1/15*(3*x^8 + 12*x^7 + 18*x^6 + 12*x^5 + 2*x^4 - 2*x^3 - x^2 - 2)*sqrt(-x^4 - 2*
x^3 - x^2 + 1)

_______________________________________________________________________________________

Sympy [A]  time = 2.15936, size = 182, normalized size = 4.33 \[ \frac{x^{8} \sqrt{- x^{4} - 2 x^{3} - x^{2} + 1}}{5} + \frac{4 x^{7} \sqrt{- x^{4} - 2 x^{3} - x^{2} + 1}}{5} + \frac{6 x^{6} \sqrt{- x^{4} - 2 x^{3} - x^{2} + 1}}{5} + \frac{4 x^{5} \sqrt{- x^{4} - 2 x^{3} - x^{2} + 1}}{5} + \frac{2 x^{4} \sqrt{- x^{4} - 2 x^{3} - x^{2} + 1}}{15} - \frac{2 x^{3} \sqrt{- x^{4} - 2 x^{3} - x^{2} + 1}}{15} - \frac{x^{2} \sqrt{- x^{4} - 2 x^{3} - x^{2} + 1}}{15} - \frac{2 \sqrt{- x^{4} - 2 x^{3} - x^{2} + 1}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**8*sqrt(-x**4 - 2*x**3 - x**2 + 1)/5 + 4*x**7*sqrt(-x**4 - 2*x**3 - x**2 + 1)/
5 + 6*x**6*sqrt(-x**4 - 2*x**3 - x**2 + 1)/5 + 4*x**5*sqrt(-x**4 - 2*x**3 - x**2
+ 1)/5 + 2*x**4*sqrt(-x**4 - 2*x**3 - x**2 + 1)/15 - 2*x**3*sqrt(-x**4 - 2*x**3
- x**2 + 1)/15 - x**2*sqrt(-x**4 - 2*x**3 - x**2 + 1)/15 - 2*sqrt(-x**4 - 2*x**
3 - x**2 + 1)/15

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.26467, size = 69, normalized size = 1.64 \[ \frac{1}{15} \, \sqrt{-x^{4} - 2 \, x^{3} - x^{2} + 1}{\left ({\left ({\left ({\left (3 \,{\left ({\left ({\left (x + 4\right )} x + 6\right )} x + 4\right )} x + 2\right )} x - 2\right )} x - 1\right )} x^{2} - 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^4 - 2*x^3 - x^2 + 1)*(2*x + 1)*(x + 1)^3*x^3,x, algorithm="giac")

[Out]

1/15*sqrt(-x^4 - 2*x^3 - x^2 + 1)*((((3*(((x + 4)*x + 6)*x + 4)*x + 2)*x - 2)*x
- 1)*x^2 - 2)