3.1089 \(\int (1-x)^{9/2} (1+x)^{5/2} \, dx\)

Optimal. Leaf size=110 \[ \frac{1}{8} (x+1)^{7/2} (1-x)^{9/2}+\frac{9}{56} (x+1)^{7/2} (1-x)^{7/2}+\frac{3}{16} x (x+1)^{5/2} (1-x)^{5/2}+\frac{15}{64} x (x+1)^{3/2} (1-x)^{3/2}+\frac{45}{128} x \sqrt{x+1} \sqrt{1-x}+\frac{45}{128} \sin ^{-1}(x) \]

[Out]

(45*Sqrt[1 - x]*x*Sqrt[1 + x])/128 + (15*(1 - x)^(3/2)*x*(1 + x)^(3/2))/64 + (3*
(1 - x)^(5/2)*x*(1 + x)^(5/2))/16 + (9*(1 - x)^(7/2)*(1 + x)^(7/2))/56 + ((1 - x
)^(9/2)*(1 + x)^(7/2))/8 + (45*ArcSin[x])/128

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Rubi [A]  time = 0.0703303, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \frac{1}{8} (x+1)^{7/2} (1-x)^{9/2}+\frac{9}{56} (x+1)^{7/2} (1-x)^{7/2}+\frac{3}{16} x (x+1)^{5/2} (1-x)^{5/2}+\frac{15}{64} x (x+1)^{3/2} (1-x)^{3/2}+\frac{45}{128} x \sqrt{x+1} \sqrt{1-x}+\frac{45}{128} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(1 - x)^(9/2)*(1 + x)^(5/2),x]

[Out]

(45*Sqrt[1 - x]*x*Sqrt[1 + x])/128 + (15*(1 - x)^(3/2)*x*(1 + x)^(3/2))/64 + (3*
(1 - x)^(5/2)*x*(1 + x)^(5/2))/16 + (9*(1 - x)^(7/2)*(1 + x)^(7/2))/56 + ((1 - x
)^(9/2)*(1 + x)^(7/2))/8 + (45*ArcSin[x])/128

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Rubi in Sympy [A]  time = 9.64029, size = 94, normalized size = 0.85 \[ \frac{3 x \left (- x + 1\right )^{\frac{5}{2}} \left (x + 1\right )^{\frac{5}{2}}}{16} + \frac{15 x \left (- x + 1\right )^{\frac{3}{2}} \left (x + 1\right )^{\frac{3}{2}}}{64} + \frac{45 x \sqrt{- x + 1} \sqrt{x + 1}}{128} + \frac{\left (- x + 1\right )^{\frac{9}{2}} \left (x + 1\right )^{\frac{7}{2}}}{8} + \frac{9 \left (- x + 1\right )^{\frac{7}{2}} \left (x + 1\right )^{\frac{7}{2}}}{56} + \frac{45 \operatorname{asin}{\left (x \right )}}{128} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-x)**(9/2)*(1+x)**(5/2),x)

[Out]

3*x*(-x + 1)**(5/2)*(x + 1)**(5/2)/16 + 15*x*(-x + 1)**(3/2)*(x + 1)**(3/2)/64 +
45*x*sqrt(-x + 1)*sqrt(x + 1)/128 + (-x + 1)**(9/2)*(x + 1)**(7/2)/8 + 9*(-x +
1)**(7/2)*(x + 1)**(7/2)/56 + 45*asin(x)/128

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Mathematica [A]  time = 0.061726, size = 68, normalized size = 0.62 \[ \frac{1}{896} \left (\sqrt{1-x^2} \left (112 x^7-256 x^6-168 x^5+768 x^4-210 x^3-768 x^2+581 x+256\right )+630 \sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - x)^(9/2)*(1 + x)^(5/2),x]

[Out]

(Sqrt[1 - x^2]*(256 + 581*x - 768*x^2 - 210*x^3 + 768*x^4 - 168*x^5 - 256*x^6 +
112*x^7) + 630*ArcSin[Sqrt[1 + x]/Sqrt[2]])/896

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Maple [A]  time = 0.007, size = 141, normalized size = 1.3 \[{\frac{1}{8} \left ( 1-x \right ) ^{{\frac{9}{2}}} \left ( 1+x \right ) ^{{\frac{7}{2}}}}+{\frac{9}{56} \left ( 1-x \right ) ^{{\frac{7}{2}}} \left ( 1+x \right ) ^{{\frac{7}{2}}}}+{\frac{3}{16} \left ( 1-x \right ) ^{{\frac{5}{2}}} \left ( 1+x \right ) ^{{\frac{7}{2}}}}+{\frac{3}{16} \left ( 1-x \right ) ^{{\frac{3}{2}}} \left ( 1+x \right ) ^{{\frac{7}{2}}}}+{\frac{9}{64}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{7}{2}}}}-{\frac{3}{64}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{5}{2}}}}-{\frac{15}{128}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{3}{2}}}}-{\frac{45}{128}\sqrt{1-x}\sqrt{1+x}}+{\frac{45\,\arcsin \left ( x \right ) }{128}\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-x)^(9/2)*(1+x)^(5/2),x)

[Out]

1/8*(1-x)^(9/2)*(1+x)^(7/2)+9/56*(1-x)^(7/2)*(1+x)^(7/2)+3/16*(1-x)^(5/2)*(1+x)^
(7/2)+3/16*(1-x)^(3/2)*(1+x)^(7/2)+9/64*(1-x)^(1/2)*(1+x)^(7/2)-3/64*(1-x)^(1/2)
*(1+x)^(5/2)-15/128*(1-x)^(1/2)*(1+x)^(3/2)-45/128*(1-x)^(1/2)*(1+x)^(1/2)+45/12
8*((1+x)*(1-x))^(1/2)/(1+x)^(1/2)/(1-x)^(1/2)*arcsin(x)

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Maxima [A]  time = 1.49644, size = 86, normalized size = 0.78 \[ -\frac{1}{8} \,{\left (-x^{2} + 1\right )}^{\frac{7}{2}} x + \frac{2}{7} \,{\left (-x^{2} + 1\right )}^{\frac{7}{2}} + \frac{3}{16} \,{\left (-x^{2} + 1\right )}^{\frac{5}{2}} x + \frac{15}{64} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} x + \frac{45}{128} \, \sqrt{-x^{2} + 1} x + \frac{45}{128} \, \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^(5/2)*(-x + 1)^(9/2),x, algorithm="maxima")

[Out]

-1/8*(-x^2 + 1)^(7/2)*x + 2/7*(-x^2 + 1)^(7/2) + 3/16*(-x^2 + 1)^(5/2)*x + 15/64
*(-x^2 + 1)^(3/2)*x + 45/128*sqrt(-x^2 + 1)*x + 45/128*arcsin(x)

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Fricas [A]  time = 0.216722, size = 386, normalized size = 3.51 \[ -\frac{896 \, x^{15} - 2048 \, x^{14} - 11200 \, x^{13} + 28672 \, x^{12} + 43568 \, x^{11} - 143360 \, x^{10} - 58408 \, x^{9} + 360192 \, x^{8} - 40152 \, x^{7} - 501760 \, x^{6} + 203728 \, x^{5} + 372736 \, x^{4} - 212800 \, x^{3} - 114688 \, x^{2} -{\left (112 \, x^{15} - 256 \, x^{14} - 3752 \, x^{13} + 8960 \, x^{12} + 23086 \, x^{11} - 66304 \, x^{10} - 48251 \, x^{9} + 213248 \, x^{8} + 5152 \, x^{7} - 358400 \, x^{6} + 125216 \, x^{5} + 315392 \, x^{4} - 175616 \, x^{3} - 114688 \, x^{2} + 74368 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} + 630 \,{\left (x^{8} - 32 \, x^{6} + 160 \, x^{4} - 256 \, x^{2} + 8 \,{\left (x^{6} - 10 \, x^{4} + 24 \, x^{2} - 16\right )} \sqrt{x + 1} \sqrt{-x + 1} + 128\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) + 74368 \, x}{896 \,{\left (x^{8} - 32 \, x^{6} + 160 \, x^{4} - 256 \, x^{2} + 8 \,{\left (x^{6} - 10 \, x^{4} + 24 \, x^{2} - 16\right )} \sqrt{x + 1} \sqrt{-x + 1} + 128\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^(5/2)*(-x + 1)^(9/2),x, algorithm="fricas")

[Out]

-1/896*(896*x^15 - 2048*x^14 - 11200*x^13 + 28672*x^12 + 43568*x^11 - 143360*x^1
0 - 58408*x^9 + 360192*x^8 - 40152*x^7 - 501760*x^6 + 203728*x^5 + 372736*x^4 -
212800*x^3 - 114688*x^2 - (112*x^15 - 256*x^14 - 3752*x^13 + 8960*x^12 + 23086*x
^11 - 66304*x^10 - 48251*x^9 + 213248*x^8 + 5152*x^7 - 358400*x^6 + 125216*x^5 +
315392*x^4 - 175616*x^3 - 114688*x^2 + 74368*x)*sqrt(x + 1)*sqrt(-x + 1) + 630*
(x^8 - 32*x^6 + 160*x^4 - 256*x^2 + 8*(x^6 - 10*x^4 + 24*x^2 - 16)*sqrt(x + 1)*s
qrt(-x + 1) + 128)*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x) + 74368*x)/(x^8 - 32
*x^6 + 160*x^4 - 256*x^2 + 8*(x^6 - 10*x^4 + 24*x^2 - 16)*sqrt(x + 1)*sqrt(-x +
1) + 128)

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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)**(9/2)*(1+x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.246424, size = 335, normalized size = 3.05 \[ -\frac{2}{105} \,{\left ({\left (3 \,{\left ({\left (5 \,{\left (x + 1\right )}{\left (x - 5\right )} + 74\right )}{\left (x + 1\right )} - 96\right )}{\left (x + 1\right )} + 203\right )}{\left (x + 1\right )} - 70\right )}{\left (x + 1\right )}^{\frac{3}{2}} \sqrt{-x + 1} + \frac{4}{15} \,{\left ({\left (3 \,{\left (x + 1\right )}{\left (x - 3\right )} + 17\right )}{\left (x + 1\right )} - 10\right )}{\left (x + 1\right )}^{\frac{3}{2}} \sqrt{-x + 1} - \frac{2}{3} \,{\left (x + 1\right )}^{\frac{3}{2}}{\left (x - 1\right )} \sqrt{-x + 1} + \frac{1}{384} \,{\left ({\left (2 \,{\left ({\left (4 \,{\left ({\left (6 \,{\left (x + 1\right )}{\left (x - 6\right )} + 125\right )}{\left (x + 1\right )} - 205\right )}{\left (x + 1\right )} + 795\right )}{\left (x + 1\right )} - 449\right )}{\left (x + 1\right )} + 251\right )}{\left (x + 1\right )} - 15\right )} \sqrt{x + 1} \sqrt{-x + 1} - \frac{1}{48} \,{\left ({\left (2 \,{\left ({\left (4 \,{\left (x + 1\right )}{\left (x - 4\right )} + 39\right )}{\left (x + 1\right )} - 37\right )}{\left (x + 1\right )} + 31\right )}{\left (x + 1\right )} - 3\right )} \sqrt{x + 1} \sqrt{-x + 1} - \frac{1}{8} \,{\left ({\left (2 \,{\left (x + 1\right )}{\left (x - 2\right )} + 5\right )}{\left (x + 1\right )} - 1\right )} \sqrt{x + 1} \sqrt{-x + 1} + \frac{1}{2} \, \sqrt{x + 1} x \sqrt{-x + 1} + \frac{45}{64} \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^(5/2)*(-x + 1)^(9/2),x, algorithm="giac")

[Out]

-2/105*((3*((5*(x + 1)*(x - 5) + 74)*(x + 1) - 96)*(x + 1) + 203)*(x + 1) - 70)*
(x + 1)^(3/2)*sqrt(-x + 1) + 4/15*((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) + 1/384*((2*((4*((
6*(x + 1)*(x - 6) + 125)*(x + 1) - 205)*(x + 1) + 795)*(x + 1) - 449)*(x + 1) +
251)*(x + 1) - 15)*sqrt(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) - 1/8*((2*(x
+ 1)*(x - 2) + 5)*(x + 1) - 1)*sqrt(x + 1)*sqrt(-x + 1) + 1/2*sqrt(x + 1)*x*sqrt
(-x + 1) + 45/64*arcsin(1/2*sqrt(2)*sqrt(x + 1))