3.1091 \(\int (1-x)^{5/2} (1+x)^{5/2} \, dx\)

Optimal. Leaf size=70 \[ \frac{1}{6} (1-x)^{5/2} x (x+1)^{5/2}+\frac{5}{24} (1-x)^{3/2} x (x+1)^{3/2}+\frac{5}{16} \sqrt{1-x} x \sqrt{x+1}+\frac{5}{16} \sin ^{-1}(x) \]

[Out]

(5*Sqrt[1 - x]*x*Sqrt[1 + x])/16 + (5*(1 - x)^(3/2)*x*(1 + x)^(3/2))/24 + ((1 -
x)^(5/2)*x*(1 + x)^(5/2))/6 + (5*ArcSin[x])/16

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Rubi [A]  time = 0.0450069, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{1}{6} (1-x)^{5/2} x (x+1)^{5/2}+\frac{5}{24} (1-x)^{3/2} x (x+1)^{3/2}+\frac{5}{16} \sqrt{1-x} x \sqrt{x+1}+\frac{5}{16} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(1 - x)^(5/2)*(1 + x)^(5/2),x]

[Out]

(5*Sqrt[1 - x]*x*Sqrt[1 + x])/16 + (5*(1 - x)^(3/2)*x*(1 + x)^(3/2))/24 + ((1 -
x)^(5/2)*x*(1 + x)^(5/2))/6 + (5*ArcSin[x])/16

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Rubi in Sympy [A]  time = 6.88058, size = 60, normalized size = 0.86 \[ \frac{x \left (- x + 1\right )^{\frac{5}{2}} \left (x + 1\right )^{\frac{5}{2}}}{6} + \frac{5 x \left (- x + 1\right )^{\frac{3}{2}} \left (x + 1\right )^{\frac{3}{2}}}{24} + \frac{5 x \sqrt{- x + 1} \sqrt{x + 1}}{16} + \frac{5 \operatorname{asin}{\left (x \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-x)**(5/2)*(1+x)**(5/2),x)

[Out]

x*(-x + 1)**(5/2)*(x + 1)**(5/2)/6 + 5*x*(-x + 1)**(3/2)*(x + 1)**(3/2)/24 + 5*x
*sqrt(-x + 1)*sqrt(x + 1)/16 + 5*asin(x)/16

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Mathematica [A]  time = 0.0293732, size = 34, normalized size = 0.49 \[ \frac{1}{48} \left (x \sqrt{1-x^2} \left (8 x^4-26 x^2+33\right )+15 \sin ^{-1}(x)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - x)^(5/2)*(1 + x)^(5/2),x]

[Out]

(x*Sqrt[1 - x^2]*(33 - 26*x^2 + 8*x^4) + 15*ArcSin[x])/48

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Maple [B]  time = 0.004, size = 113, normalized size = 1.6 \[{\frac{1}{6} \left ( 1-x \right ) ^{{\frac{5}{2}}} \left ( 1+x \right ) ^{{\frac{7}{2}}}}+{\frac{1}{6} \left ( 1-x \right ) ^{{\frac{3}{2}}} \left ( 1+x \right ) ^{{\frac{7}{2}}}}+{\frac{1}{8}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{7}{2}}}}-{\frac{1}{24}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{5}{2}}}}-{\frac{5}{48}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{3}{2}}}}-{\frac{5}{16}\sqrt{1-x}\sqrt{1+x}}+{\frac{5\,\arcsin \left ( x \right ) }{16}\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-x)^(5/2)*(1+x)^(5/2),x)

[Out]

1/6*(1-x)^(5/2)*(1+x)^(7/2)+1/6*(1-x)^(3/2)*(1+x)^(7/2)+1/8*(1-x)^(1/2)*(1+x)^(7
/2)-1/24*(1-x)^(1/2)*(1+x)^(5/2)-5/48*(1-x)^(1/2)*(1+x)^(3/2)-5/16*(1-x)^(1/2)*(
1+x)^(1/2)+5/16*((1+x)*(1-x))^(1/2)/(1+x)^(1/2)/(1-x)^(1/2)*arcsin(x)

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Maxima [A]  time = 1.49399, size = 55, normalized size = 0.79 \[ \frac{1}{6} \,{\left (-x^{2} + 1\right )}^{\frac{5}{2}} x + \frac{5}{24} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} x + \frac{5}{16} \, \sqrt{-x^{2} + 1} x + \frac{5}{16} \, \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^(5/2)*(-x + 1)^(5/2),x, algorithm="maxima")

[Out]

1/6*(-x^2 + 1)^(5/2)*x + 5/24*(-x^2 + 1)^(3/2)*x + 5/16*sqrt(-x^2 + 1)*x + 5/16*
arcsin(x)

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Fricas [A]  time = 0.207443, size = 243, normalized size = 3.47 \[ -\frac{48 \, x^{11} - 460 \, x^{9} + 1698 \, x^{7} - 3174 \, x^{5} + 2944 \, x^{3} -{\left (8 \, x^{11} - 170 \, x^{9} + 885 \, x^{7} - 2098 \, x^{5} + 2416 \, x^{3} - 1056 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} + 30 \,{\left (x^{6} - 18 \, x^{4} + 48 \, x^{2} + 2 \,{\left (3 \, x^{4} - 16 \, x^{2} + 16\right )} \sqrt{x + 1} \sqrt{-x + 1} - 32\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) - 1056 \, x}{48 \,{\left (x^{6} - 18 \, x^{4} + 48 \, x^{2} + 2 \,{\left (3 \, x^{4} - 16 \, x^{2} + 16\right )} \sqrt{x + 1} \sqrt{-x + 1} - 32\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^(5/2)*(-x + 1)^(5/2),x, algorithm="fricas")

[Out]

-1/48*(48*x^11 - 460*x^9 + 1698*x^7 - 3174*x^5 + 2944*x^3 - (8*x^11 - 170*x^9 +
885*x^7 - 2098*x^5 + 2416*x^3 - 1056*x)*sqrt(x + 1)*sqrt(-x + 1) + 30*(x^6 - 18*
x^4 + 48*x^2 + 2*(3*x^4 - 16*x^2 + 16)*sqrt(x + 1)*sqrt(-x + 1) - 32)*arctan((sq
rt(x + 1)*sqrt(-x + 1) - 1)/x) - 1056*x)/(x^6 - 18*x^4 + 48*x^2 + 2*(3*x^4 - 16*
x^2 + 16)*sqrt(x + 1)*sqrt(-x + 1) - 32)

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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((1-x)**(5/2)*(1+x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.224703, size = 138, normalized size = 1.97 \[ \frac{1}{48} \,{\left ({\left (2 \,{\left ({\left (4 \,{\left (x + 1\right )}{\left (x - 4\right )} + 39\right )}{\left (x + 1\right )} - 37\right )}{\left (x + 1\right )} + 31\right )}{\left (x + 1\right )} - 3\right )} \sqrt{x + 1} \sqrt{-x + 1} - \frac{1}{4} \,{\left ({\left (2 \,{\left (x + 1\right )}{\left (x - 2\right )} + 5\right )}{\left (x + 1\right )} - 1\right )} \sqrt{x + 1} \sqrt{-x + 1} + \frac{1}{2} \, \sqrt{x + 1} x \sqrt{-x + 1} + \frac{5}{8} \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^(5/2)*(-x + 1)^(5/2),x, algorithm="giac")

[Out]

1/48*((2*((4*(x + 1)*(x - 4) + 39)*(x + 1) - 37)*(x + 1) + 31)*(x + 1) - 3)*sqrt
(x + 1)*sqrt(-x + 1) - 1/4*((2*(x + 1)*(x - 2) + 5)*(x + 1) - 1)*sqrt(x + 1)*sqr
t(-x + 1) + 1/2*sqrt(x + 1)*x*sqrt(-x + 1) + 5/8*arcsin(1/2*sqrt(2)*sqrt(x + 1))