3.1076 \(\int (1-x)^{7/2} (1+x)^{3/2} \, dx\)

Optimal. Leaf size=89 \[ \frac{1}{6} (x+1)^{5/2} (1-x)^{7/2}+\frac{7}{30} (x+1)^{5/2} (1-x)^{5/2}+\frac{7}{24} x (x+1)^{3/2} (1-x)^{3/2}+\frac{7}{16} x \sqrt{x+1} \sqrt{1-x}+\frac{7}{16} \sin ^{-1}(x) \]

[Out]

(7*Sqrt[1 - x]*x*Sqrt[1 + x])/16 + (7*(1 - x)^(3/2)*x*(1 + x)^(3/2))/24 + (7*(1
- x)^(5/2)*(1 + x)^(5/2))/30 + ((1 - x)^(7/2)*(1 + x)^(5/2))/6 + (7*ArcSin[x])/1
6

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Rubi [A]  time = 0.0549936, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \frac{1}{6} (x+1)^{5/2} (1-x)^{7/2}+\frac{7}{30} (x+1)^{5/2} (1-x)^{5/2}+\frac{7}{24} x (x+1)^{3/2} (1-x)^{3/2}+\frac{7}{16} x \sqrt{x+1} \sqrt{1-x}+\frac{7}{16} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(1 - x)^(7/2)*(1 + x)^(3/2),x]

[Out]

(7*Sqrt[1 - x]*x*Sqrt[1 + x])/16 + (7*(1 - x)^(3/2)*x*(1 + x)^(3/2))/24 + (7*(1
- x)^(5/2)*(1 + x)^(5/2))/30 + ((1 - x)^(7/2)*(1 + x)^(5/2))/6 + (7*ArcSin[x])/1
6

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Rubi in Sympy [A]  time = 8.6962, size = 75, normalized size = 0.84 \[ \frac{7 x \left (- x + 1\right )^{\frac{3}{2}} \left (x + 1\right )^{\frac{3}{2}}}{24} + \frac{7 x \sqrt{- x + 1} \sqrt{x + 1}}{16} + \frac{\left (- x + 1\right )^{\frac{7}{2}} \left (x + 1\right )^{\frac{5}{2}}}{6} + \frac{7 \left (- x + 1\right )^{\frac{5}{2}} \left (x + 1\right )^{\frac{5}{2}}}{30} + \frac{7 \operatorname{asin}{\left (x \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-x)**(7/2)*(1+x)**(3/2),x)

[Out]

7*x*(-x + 1)**(3/2)*(x + 1)**(3/2)/24 + 7*x*sqrt(-x + 1)*sqrt(x + 1)/16 + (-x +
1)**(7/2)*(x + 1)**(5/2)/6 + 7*(-x + 1)**(5/2)*(x + 1)**(5/2)/30 + 7*asin(x)/16

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Mathematica [A]  time = 0.041522, size = 59, normalized size = 0.66 \[ \frac{1}{240} \sqrt{1-x^2} \left (-40 x^5+96 x^4+10 x^3-192 x^2+135 x+96\right )+\frac{7}{8} \sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - x)^(7/2)*(1 + x)^(3/2),x]

[Out]

(Sqrt[1 - x^2]*(96 + 135*x - 192*x^2 + 10*x^3 + 96*x^4 - 40*x^5))/240 + (7*ArcSi
n[Sqrt[1 + x]/Sqrt[2]])/8

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Maple [A]  time = 0.007, size = 113, normalized size = 1.3 \[{\frac{1}{6} \left ( 1-x \right ) ^{{\frac{7}{2}}} \left ( 1+x \right ) ^{{\frac{5}{2}}}}+{\frac{7}{30} \left ( 1-x \right ) ^{{\frac{5}{2}}} \left ( 1+x \right ) ^{{\frac{5}{2}}}}+{\frac{7}{24} \left ( 1-x \right ) ^{{\frac{3}{2}}} \left ( 1+x \right ) ^{{\frac{5}{2}}}}+{\frac{7}{24}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{5}{2}}}}-{\frac{7}{48}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{3}{2}}}}-{\frac{7}{16}\sqrt{1-x}\sqrt{1+x}}+{\frac{7\,\arcsin \left ( x \right ) }{16}\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-x)^(7/2)*(1+x)^(3/2),x)

[Out]

1/6*(1-x)^(7/2)*(1+x)^(5/2)+7/30*(1-x)^(5/2)*(1+x)^(5/2)+7/24*(1-x)^(3/2)*(1+x)^
(5/2)+7/24*(1-x)^(1/2)*(1+x)^(5/2)-7/48*(1-x)^(1/2)*(1+x)^(3/2)-7/16*(1-x)^(1/2)
*(1+x)^(1/2)+7/16*((1+x)*(1-x))^(1/2)/(1+x)^(1/2)/(1-x)^(1/2)*arcsin(x)

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Maxima [A]  time = 1.49403, size = 70, normalized size = 0.79 \[ -\frac{1}{6} \,{\left (-x^{2} + 1\right )}^{\frac{5}{2}} x + \frac{2}{5} \,{\left (-x^{2} + 1\right )}^{\frac{5}{2}} + \frac{7}{24} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} x + \frac{7}{16} \, \sqrt{-x^{2} + 1} x + \frac{7}{16} \, \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^(3/2)*(-x + 1)^(7/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.217779, size = 311, normalized size = 3.49 \[ \frac{240 \, x^{11} - 576 \, x^{10} - 1580 \, x^{9} + 4800 \, x^{8} + 2130 \, x^{7} - 13920 \, x^{6} + 3210 \, x^{5} + 17280 \, x^{4} - 8320 \, x^{3} - 7680 \, x^{2} -{\left (40 \, x^{11} - 96 \, x^{10} - 730 \, x^{9} + 1920 \, x^{8} + 1965 \, x^{7} - 8160 \, x^{6} + 670 \, x^{5} + 13440 \, x^{4} - 6160 \, x^{3} - 7680 \, x^{2} + 4320 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} - 210 \,{\left (x^{6} - 18 \, x^{4} + 48 \, x^{2} + 2 \,{\left (3 \, x^{4} - 16 \, x^{2} + 16\right )} \sqrt{x + 1} \sqrt{-x + 1} - 32\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) + 4320 \, x}{240 \,{\left (x^{6} - 18 \, x^{4} + 48 \, x^{2} + 2 \,{\left (3 \, x^{4} - 16 \, x^{2} + 16\right )} \sqrt{x + 1} \sqrt{-x + 1} - 32\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^(3/2)*(-x + 1)^(7/2),x, algorithm="fricas")

[Out]

1/240*(240*x^11 - 576*x^10 - 1580*x^9 + 4800*x^8 + 2130*x^7 - 13920*x^6 + 3210*x
^5 + 17280*x^4 - 8320*x^3 - 7680*x^2 - (40*x^11 - 96*x^10 - 730*x^9 + 1920*x^8 +
1965*x^7 - 8160*x^6 + 670*x^5 + 13440*x^4 - 6160*x^3 - 7680*x^2 + 4320*x)*sqrt(
x + 1)*sqrt(-x + 1) - 210*(x^6 - 18*x^4 + 48*x^2 + 2*(3*x^4 - 16*x^2 + 16)*sqrt(
x + 1)*sqrt(-x + 1) - 32)*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x) + 4320*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)**(7/2)*(1+x)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.227891, size = 161, normalized size = 1.81 \[ \frac{2}{15} \,{\left ({\left (3 \,{\left (x + 1\right )}{\left (x - 3\right )} + 17\right )}{\left (x + 1\right )} - 10\right )}{\left (x + 1\right )}^{\frac{3}{2}} \sqrt{-x + 1} - \frac{2}{3} \,{\left (x + 1\right )}^{\frac{3}{2}}{\left (x - 1\right )} \sqrt{-x + 1} - \frac{1}{48} \,{\left ({\left (2 \,{\left ({\left (4 \,{\left (x + 1\right )}{\left (x - 4\right )} + 39\right )}{\left (x + 1\right )} - 37\right )}{\left (x + 1\right )} + 31\right )}{\left (x + 1\right )} - 3\right )} \sqrt{x + 1} \sqrt{-x + 1} + \frac{1}{2} \, \sqrt{x + 1} x \sqrt{-x + 1} + \frac{7}{8} \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^(3/2)*(-x + 1)^(7/2),x, algorithm="giac")

[Out]

2/15*((3*(x + 1)*(x - 3) + 17)*(x + 1) - 10)*(x + 1)^(3/2)*sqrt(-x + 1) - 2/3*(x
+ 1)^(3/2)*(x - 1)*sqrt(-x + 1) - 1/48*((2*((4*(x + 1)*(x - 4) + 39)*(x + 1) -
37)*(x + 1) + 31)*(x + 1) - 3)*sqrt(x + 1)*sqrt(-x + 1) + 1/2*sqrt(x + 1)*x*sqrt
(-x + 1) + 7/8*arcsin(1/2*sqrt(2)*sqrt(x + 1))