3.1077 \(\int (1-x)^{5/2} (1+x)^{3/2} \, dx\)

Optimal. Leaf size=69 \[ \frac{1}{5} (1-x)^{5/2} (x+1)^{5/2}+\frac{1}{4} (1-x)^{3/2} x (x+1)^{3/2}+\frac{3}{8} \sqrt{1-x} x \sqrt{x+1}+\frac{3}{8} \sin ^{-1}(x) \]

[Out]

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

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Rubi [A]  time = 0.0402593, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \frac{1}{5} (1-x)^{5/2} (x+1)^{5/2}+\frac{1}{4} (1-x)^{3/2} x (x+1)^{3/2}+\frac{3}{8} \sqrt{1-x} x \sqrt{x+1}+\frac{3}{8} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(1 - x)^(5/2)*(1 + x)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 6.65919, size = 56, normalized size = 0.81 \[ \frac{x \left (- x + 1\right )^{\frac{3}{2}} \left (x + 1\right )^{\frac{3}{2}}}{4} + \frac{3 x \sqrt{- x + 1} \sqrt{x + 1}}{8} + \frac{\left (- x + 1\right )^{\frac{5}{2}} \left (x + 1\right )^{\frac{5}{2}}}{5} + \frac{3 \operatorname{asin}{\left (x \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-x)**(5/2)*(1+x)**(3/2),x)

[Out]

x*(-x + 1)**(3/2)*(x + 1)**(3/2)/4 + 3*x*sqrt(-x + 1)*sqrt(x + 1)/8 + (-x + 1)**
(5/2)*(x + 1)**(5/2)/5 + 3*asin(x)/8

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Mathematica [A]  time = 0.0336619, size = 54, normalized size = 0.78 \[ \frac{1}{40} \sqrt{1-x^2} \left (8 x^4-10 x^3-16 x^2+25 x+8\right )+\frac{3}{4} \sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - x)^(5/2)*(1 + x)^(3/2),x]

[Out]

(Sqrt[1 - x^2]*(8 + 25*x - 16*x^2 - 10*x^3 + 8*x^4))/40 + (3*ArcSin[Sqrt[1 + x]/
Sqrt[2]])/4

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Maple [A]  time = 0.007, size = 99, normalized size = 1.4 \[{\frac{1}{5} \left ( 1-x \right ) ^{{\frac{5}{2}}} \left ( 1+x \right ) ^{{\frac{5}{2}}}}+{\frac{1}{4} \left ( 1-x \right ) ^{{\frac{3}{2}}} \left ( 1+x \right ) ^{{\frac{5}{2}}}}+{\frac{1}{4}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{5}{2}}}}-{\frac{1}{8}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{3}{2}}}}-{\frac{3}{8}\sqrt{1-x}\sqrt{1+x}}+{\frac{3\,\arcsin \left ( x \right ) }{8}\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-x)^(5/2)*(1+x)^(3/2),x)

[Out]

1/5*(1-x)^(5/2)*(1+x)^(5/2)+1/4*(1-x)^(3/2)*(1+x)^(5/2)+1/4*(1-x)^(1/2)*(1+x)^(5
/2)-1/8*(1-x)^(1/2)*(1+x)^(3/2)-3/8*(1-x)^(1/2)*(1+x)^(1/2)+3/8*((1+x)*(1-x))^(1
/2)/(1+x)^(1/2)/(1-x)^(1/2)*arcsin(x)

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Maxima [A]  time = 1.48145, size = 54, normalized size = 0.78 \[ \frac{1}{5} \,{\left (-x^{2} + 1\right )}^{\frac{5}{2}} + \frac{1}{4} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} x + \frac{3}{8} \, \sqrt{-x^{2} + 1} x + \frac{3}{8} \, \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^(3/2)*(-x + 1)^(5/2),x, algorithm="maxima")

[Out]

1/5*(-x^2 + 1)^(5/2) + 1/4*(-x^2 + 1)^(3/2)*x + 3/8*sqrt(-x^2 + 1)*x + 3/8*arcsi
n(x)

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Fricas [A]  time = 0.217072, size = 270, normalized size = 3.91 \[ \frac{8 \, x^{10} - 10 \, x^{9} - 120 \, x^{8} + 155 \, x^{7} + 440 \, x^{6} - 605 \, x^{5} - 640 \, x^{4} + 860 \, x^{3} + 320 \, x^{2} + 5 \,{\left (8 \, x^{8} - 10 \, x^{7} - 48 \, x^{6} + 65 \, x^{5} + 96 \, x^{4} - 132 \, x^{3} - 64 \, x^{2} + 80 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} - 30 \,{\left (5 \, x^{4} - 20 \, x^{2} -{\left (x^{4} - 12 \, x^{2} + 16\right )} \sqrt{x + 1} \sqrt{-x + 1} + 16\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) - 400 \, x}{40 \,{\left (5 \, x^{4} - 20 \, x^{2} -{\left (x^{4} - 12 \, x^{2} + 16\right )} \sqrt{x + 1} \sqrt{-x + 1} + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^(3/2)*(-x + 1)^(5/2),x, algorithm="fricas")

[Out]

1/40*(8*x^10 - 10*x^9 - 120*x^8 + 155*x^7 + 440*x^6 - 605*x^5 - 640*x^4 + 860*x^
3 + 320*x^2 + 5*(8*x^8 - 10*x^7 - 48*x^6 + 65*x^5 + 96*x^4 - 132*x^3 - 64*x^2 +
80*x)*sqrt(x + 1)*sqrt(-x + 1) - 30*(5*x^4 - 20*x^2 - (x^4 - 12*x^2 + 16)*sqrt(x
+ 1)*sqrt(-x + 1) + 16)*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x) - 400*x)/(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((1-x)**(5/2)*(1+x)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.225104, size = 143, normalized size = 2.07 \[ \frac{1}{15} \,{\left ({\left (3 \,{\left (x + 1\right )}{\left (x - 3\right )} + 17\right )}{\left (x + 1\right )} - 10\right )}{\left (x + 1\right )}^{\frac{3}{2}} \sqrt{-x + 1} - \frac{1}{3} \,{\left (x + 1\right )}^{\frac{3}{2}}{\left (x - 1\right )} \sqrt{-x + 1} - \frac{1}{8} \,{\left ({\left (2 \,{\left (x + 1\right )}{\left (x - 2\right )} + 5\right )}{\left (x + 1\right )} - 1\right )} \sqrt{x + 1} \sqrt{-x + 1} + \frac{1}{2} \, \sqrt{x + 1} x \sqrt{-x + 1} + \frac{3}{4} \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^(3/2)*(-x + 1)^(5/2),x, algorithm="giac")

[Out]

1/15*((3*(x + 1)*(x - 3) + 17)*(x + 1) - 10)*(x + 1)^(3/2)*sqrt(-x + 1) - 1/3*(x
+ 1)^(3/2)*(x - 1)*sqrt(-x + 1) - 1/8*((2*(x + 1)*(x - 2) + 5)*(x + 1) - 1)*sqr
t(x + 1)*sqrt(-x + 1) + 1/2*sqrt(x + 1)*x*sqrt(-x + 1) + 3/4*arcsin(1/2*sqrt(2)*
sqrt(x + 1))