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]
_______________________________________________________________________________________
Rubi [A] time = 0.395042, antiderivative size = 59, normalized size of antiderivative = 1.4, number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ -\frac{1}{5} x^2 \left (-x^4-2 x^3-x^2+1\right )^{3/2} (x+1)^2-\frac{2}{15} \left (-x^4-2 x^3-x^2+1\right )^{3/2} \]
Antiderivative was successfully verified.
[In] Int[(1 + 2*x)*(x + x^2)^3*Sqrt[1 - (x + x^2)^2],x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 21.1478, 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((1+2*x)*(x**2+x)**3*(1-(x**2+x)**2)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0714596, 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[(1 + 2*x)*(x + x^2)^3*Sqrt[1 - (x + x^2)^2],x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.008, size = 51, normalized size = 1.2 \[{\frac{ \left ( 3\,{x}^{4}+6\,{x}^{3}+3\,{x}^{2}+2 \right ) \left ({x}^{2}+x+1 \right ) \left ({x}^{2}+x-1 \right ) }{15}\sqrt{-{x}^{4}-2\,{x}^{3}-{x}^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1+2*x)*(x^2+x)^3*(1-(x^2+x)^2)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.876713, 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^2 + x)^2 + 1)*(x^2 + x)^3*(2*x + 1),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.260162, 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^2 + x)^2 + 1)*(x^2 + x)^3*(2*x + 1),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 40.747, 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((1+2*x)*(x**2+x)**3*(1-(x**2+x)**2)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.264102, 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^2 + x)^2 + 1)*(x^2 + x)^3*(2*x + 1),x, algorithm="giac")
[Out]