3.1063 \(\int (1-x)^{9/2} \sqrt{1+x} \, dx\)

Optimal. Leaf size=108 \[ \frac{1}{6} (x+1)^{3/2} (1-x)^{9/2}+\frac{3}{10} (x+1)^{3/2} (1-x)^{7/2}+\frac{21}{40} (x+1)^{3/2} (1-x)^{5/2}+\frac{7}{8} (x+1)^{3/2} (1-x)^{3/2}+\frac{21}{16} x \sqrt{x+1} \sqrt{1-x}+\frac{21}{16} \sin ^{-1}(x) \]

[Out]

(21*Sqrt[1 - x]*x*Sqrt[1 + x])/16 + (7*(1 - x)^(3/2)*(1 + x)^(3/2))/8 + (21*(1 -
x)^(5/2)*(1 + x)^(3/2))/40 + (3*(1 - x)^(7/2)*(1 + x)^(3/2))/10 + ((1 - x)^(9/2
)*(1 + x)^(3/2))/6 + (21*ArcSin[x])/16

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Rubi [A]  time = 0.0750559, antiderivative size = 108, 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 \[ \frac{1}{6} (x+1)^{3/2} (1-x)^{9/2}+\frac{3}{10} (x+1)^{3/2} (1-x)^{7/2}+\frac{21}{40} (x+1)^{3/2} (1-x)^{5/2}+\frac{7}{8} (x+1)^{3/2} (1-x)^{3/2}+\frac{21}{16} x \sqrt{x+1} \sqrt{1-x}+\frac{21}{16} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

(21*Sqrt[1 - x]*x*Sqrt[1 + x])/16 + (7*(1 - x)^(3/2)*(1 + x)^(3/2))/8 + (21*(1 -
x)^(5/2)*(1 + x)^(3/2))/40 + (3*(1 - x)^(7/2)*(1 + x)^(3/2))/10 + ((1 - x)^(9/2
)*(1 + x)^(3/2))/6 + (21*ArcSin[x])/16

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Rubi in Sympy [A]  time = 10.2972, size = 90, normalized size = 0.83 \[ \frac{21 x \sqrt{- x + 1} \sqrt{x + 1}}{16} + \frac{\left (- x + 1\right )^{\frac{9}{2}} \left (x + 1\right )^{\frac{3}{2}}}{6} + \frac{3 \left (- x + 1\right )^{\frac{7}{2}} \left (x + 1\right )^{\frac{3}{2}}}{10} + \frac{21 \left (- x + 1\right )^{\frac{5}{2}} \left (x + 1\right )^{\frac{3}{2}}}{40} + \frac{7 \left (- x + 1\right )^{\frac{3}{2}} \left (x + 1\right )^{\frac{3}{2}}}{8} + \frac{21 \operatorname{asin}{\left (x \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

21*x*sqrt(-x + 1)*sqrt(x + 1)/16 + (-x + 1)**(9/2)*(x + 1)**(3/2)/6 + 3*(-x + 1)
**(7/2)*(x + 1)**(3/2)/10 + 21*(-x + 1)**(5/2)*(x + 1)**(3/2)/40 + 7*(-x + 1)**(
3/2)*(x + 1)**(3/2)/8 + 21*asin(x)/16

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Mathematica [A]  time = 0.0437452, size = 59, normalized size = 0.55 \[ \frac{1}{240} \sqrt{1-x^2} \left (40 x^5-192 x^4+350 x^3-256 x^2-75 x+448\right )+\frac{21}{8} \sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - x^2]*(448 - 75*x - 256*x^2 + 350*x^3 - 192*x^4 + 40*x^5))/240 + (21*Ar
cSin[Sqrt[1 + x]/Sqrt[2]])/8

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Maple [A]  time = 0.01, size = 113, normalized size = 1.1 \[{\frac{1}{6} \left ( 1-x \right ) ^{{\frac{9}{2}}} \left ( 1+x \right ) ^{{\frac{3}{2}}}}+{\frac{3}{10} \left ( 1-x \right ) ^{{\frac{7}{2}}} \left ( 1+x \right ) ^{{\frac{3}{2}}}}+{\frac{21}{40} \left ( 1-x \right ) ^{{\frac{5}{2}}} \left ( 1+x \right ) ^{{\frac{3}{2}}}}+{\frac{7}{8} \left ( 1-x \right ) ^{{\frac{3}{2}}} \left ( 1+x \right ) ^{{\frac{3}{2}}}}+{\frac{21}{16}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{3}{2}}}}-{\frac{21}{16}\sqrt{1-x}\sqrt{1+x}}+{\frac{21\,\arcsin \left ( x \right ) }{16}\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-x)^(9/2)*(1+x)^(1/2),x)

[Out]

1/6*(1-x)^(9/2)*(1+x)^(3/2)+3/10*(1-x)^(7/2)*(1+x)^(3/2)+21/40*(1-x)^(5/2)*(1+x)
^(3/2)+7/8*(1-x)^(3/2)*(1+x)^(3/2)+21/16*(1-x)^(1/2)*(1+x)^(3/2)-21/16*(1-x)^(1/
2)*(1+x)^(1/2)+21/16*((1+x)*(1-x))^(1/2)/(1+x)^(1/2)/(1-x)^(1/2)*arcsin(x)

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Maxima [A]  time = 1.49029, size = 92, normalized size = 0.85 \[ -\frac{1}{6} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} x^{3} + \frac{4}{5} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} x^{2} - \frac{13}{8} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} x + \frac{28}{15} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} + \frac{21}{16} \, \sqrt{-x^{2} + 1} x + \frac{21}{16} \, \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(-x^2 + 1)^(3/2)*x^3 + 4/5*(-x^2 + 1)^(3/2)*x^2 - 13/8*(-x^2 + 1)^(3/2)*x +
28/15*(-x^2 + 1)^(3/2) + 21/16*sqrt(-x^2 + 1)*x + 21/16*arcsin(x)

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Fricas [A]  time = 0.210511, size = 311, normalized size = 2.88 \[ -\frac{240 \, x^{11} - 1152 \, x^{10} + 580 \, x^{9} + 5760 \, x^{8} - 11190 \, x^{7} - 320 \, x^{6} + 23970 \, x^{5} - 19200 \, x^{4} - 16000 \, x^{3} + 15360 \, x^{2} -{\left (40 \, x^{11} - 192 \, x^{10} - 370 \, x^{9} + 3200 \, x^{8} - 4455 \, x^{7} - 4160 \, x^{6} + 16870 \, x^{5} - 11520 \, x^{4} - 14800 \, x^{3} + 15360 \, x^{2} + 2400 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} + 630 \,{\left (x^{6} - 18 \, x^{4} + 48 \, x^{2} + 2 \,{\left (3 \, x^{4} - 16 \, x^{2} + 16\right )} \sqrt{x + 1} \sqrt{-x + 1} - 32\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) + 2400 \, x}{240 \,{\left (x^{6} - 18 \, x^{4} + 48 \, x^{2} + 2 \,{\left (3 \, x^{4} - 16 \, x^{2} + 16\right )} \sqrt{x + 1} \sqrt{-x + 1} - 32\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/240*(240*x^11 - 1152*x^10 + 580*x^9 + 5760*x^8 - 11190*x^7 - 320*x^6 + 23970*
x^5 - 19200*x^4 - 16000*x^3 + 15360*x^2 - (40*x^11 - 192*x^10 - 370*x^9 + 3200*x
^8 - 4455*x^7 - 4160*x^6 + 16870*x^5 - 11520*x^4 - 14800*x^3 + 15360*x^2 + 2400*
x)*sqrt(x + 1)*sqrt(-x + 1) + 630*(x^6 - 18*x^4 + 48*x^2 + 2*(3*x^4 - 16*x^2 + 1
6)*sqrt(x + 1)*sqrt(-x + 1) - 32)*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x) + 240
0*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)**(9/2)*(1+x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.23282, size = 201, normalized size = 1.86 \[ -\frac{4}{15} \,{\left ({\left (3 \,{\left (x + 1\right )}{\left (x - 3\right )} + 17\right )}{\left (x + 1\right )} - 10\right )}{\left (x + 1\right )}^{\frac{3}{2}} \sqrt{-x + 1} - \frac{4}{3} \,{\left (x + 1\right )}^{\frac{3}{2}}{\left (x - 1\right )} \sqrt{-x + 1} + \frac{1}{48} \,{\left ({\left (2 \,{\left ({\left (4 \,{\left (x + 1\right )}{\left (x - 4\right )} + 39\right )}{\left (x + 1\right )} - 37\right )}{\left (x + 1\right )} + 31\right )}{\left (x + 1\right )} - 3\right )} \sqrt{x + 1} \sqrt{-x + 1} + \frac{3}{4} \,{\left ({\left (2 \,{\left (x + 1\right )}{\left (x - 2\right )} + 5\right )}{\left (x + 1\right )} - 1\right )} \sqrt{x + 1} \sqrt{-x + 1} + \frac{1}{2} \, \sqrt{x + 1} x \sqrt{-x + 1} + \frac{21}{8} \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/15*((3*(x + 1)*(x - 3) + 17)*(x + 1) - 10)*(x + 1)^(3/2)*sqrt(-x + 1) - 4/3*(
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) + 3/4*((2*(x + 1)*(x -
2) + 5)*(x + 1) - 1)*sqrt(x + 1)*sqrt(-x + 1) + 1/2*sqrt(x + 1)*x*sqrt(-x + 1) +
21/8*arcsin(1/2*sqrt(2)*sqrt(x + 1))