3.258 \(\int x^3 \left (\sqrt{1-x}+\sqrt{1+x}\right )^2 \, dx\)

Optimal. Leaf size=38 \[ \frac{x^4}{2}+\frac{2}{5} \left (1-x^2\right )^{5/2}-\frac{2}{3} \left (1-x^2\right )^{3/2} \]

[Out]

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

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Rubi [A]  time = 0.197017, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ \frac{x^4}{2}+\frac{2}{5} \left (1-x^2\right )^{5/2}-\frac{2}{3} \left (1-x^2\right )^{3/2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.04812, size = 44, normalized size = 1.16 \[ \frac{1}{30} \left (x^2-1\right ) \left (3 \left (4 \sqrt{1-x^2}+5\right ) x^2+8 \sqrt{1-x^2}+15\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-1 + x^2)*(15 + 8*Sqrt[1 - x^2] + 3*x^2*(5 + 4*Sqrt[1 - x^2])))/30

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Maple [A]  time = 0.006, size = 33, normalized size = 0.9 \[{\frac{{x}^{4}}{2}}+{\frac{ \left ( 2\,{x}^{2}-2 \right ) \left ( 3\,{x}^{2}+2 \right ) }{15}\sqrt{1-x}\sqrt{1+x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.792452, size = 42, normalized size = 1.11 \[ \frac{1}{2} \, x^{4} - \frac{2}{5} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} x^{2} - \frac{4}{15} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.276094, size = 109, normalized size = 2.87 \[ \frac{12 \, x^{10} - 85 \, x^{8} + 80 \, x^{6} + 5 \,{\left (9 \, x^{8} - 16 \, x^{6}\right )} \sqrt{x + 1} \sqrt{-x + 1}}{30 \,{\left (5 \, x^{4} - 20 \, x^{2} -{\left (x^{4} - 12 \, x^{2} + 16\right )} \sqrt{x + 1} \sqrt{-x + 1} + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/30*(12*x^10 - 85*x^8 + 80*x^6 + 5*(9*x^8 - 16*x^6)*sqrt(x + 1)*sqrt(-x + 1))/(
5*x^4 - 20*x^2 - (x^4 - 12*x^2 + 16)*sqrt(x + 1)*sqrt(-x + 1) + 16)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.281216, size = 76, normalized size = 2. \[ \frac{1}{2} \,{\left (x + 1\right )}^{4} - 2 \,{\left (x + 1\right )}^{3} + \frac{2}{15} \,{\left ({\left (3 \,{\left (x + 1\right )}{\left (x - 3\right )} + 17\right )}{\left (x + 1\right )} - 10\right )}{\left (x + 1\right )}^{\frac{3}{2}} \sqrt{-x + 1} + 3 \,{\left (x + 1\right )}^{2} - 2 \, x - 2 \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(x + 1)^4 - 2*(x + 1)^3 + 2/15*((3*(x + 1)*(x - 3) + 17)*(x + 1) - 10)*(x +
1)^(3/2)*sqrt(-x + 1) + 3*(x + 1)^2 - 2*x - 2