3.1088 \(\int (1-x)^{11/2} (1+x)^{5/2} \, dx\)

Optimal. Leaf size=130 \[ \frac{1}{9} (x+1)^{7/2} (1-x)^{11/2}+\frac{11}{72} (x+1)^{7/2} (1-x)^{9/2}+\frac{11}{56} (x+1)^{7/2} (1-x)^{7/2}+\frac{11}{48} x (x+1)^{5/2} (1-x)^{5/2}+\frac{55}{192} x (x+1)^{3/2} (1-x)^{3/2}+\frac{55}{128} x \sqrt{x+1} \sqrt{1-x}+\frac{55}{128} \sin ^{-1}(x) \]

[Out]

(55*Sqrt[1 - x]*x*Sqrt[1 + x])/128 + (55*(1 - x)^(3/2)*x*(1 + x)^(3/2))/192 + (1
1*(1 - x)^(5/2)*x*(1 + x)^(5/2))/48 + (11*(1 - x)^(7/2)*(1 + x)^(7/2))/56 + (11*
(1 - x)^(9/2)*(1 + x)^(7/2))/72 + ((1 - x)^(11/2)*(1 + x)^(7/2))/9 + (55*ArcSin[
x])/128

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Rubi [A]  time = 0.0892417, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \frac{1}{9} (x+1)^{7/2} (1-x)^{11/2}+\frac{11}{72} (x+1)^{7/2} (1-x)^{9/2}+\frac{11}{56} (x+1)^{7/2} (1-x)^{7/2}+\frac{11}{48} x (x+1)^{5/2} (1-x)^{5/2}+\frac{55}{192} x (x+1)^{3/2} (1-x)^{3/2}+\frac{55}{128} x \sqrt{x+1} \sqrt{1-x}+\frac{55}{128} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(1 - x)^(11/2)*(1 + x)^(5/2),x]

[Out]

(55*Sqrt[1 - x]*x*Sqrt[1 + x])/128 + (55*(1 - x)^(3/2)*x*(1 + x)^(3/2))/192 + (1
1*(1 - x)^(5/2)*x*(1 + x)^(5/2))/48 + (11*(1 - x)^(7/2)*(1 + x)^(7/2))/56 + (11*
(1 - x)^(9/2)*(1 + x)^(7/2))/72 + ((1 - x)^(11/2)*(1 + x)^(7/2))/9 + (55*ArcSin[
x])/128

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Rubi in Sympy [A]  time = 11.5305, size = 110, normalized size = 0.85 \[ \frac{11 x \left (- x + 1\right )^{\frac{5}{2}} \left (x + 1\right )^{\frac{5}{2}}}{48} + \frac{55 x \left (- x + 1\right )^{\frac{3}{2}} \left (x + 1\right )^{\frac{3}{2}}}{192} + \frac{55 x \sqrt{- x + 1} \sqrt{x + 1}}{128} + \frac{\left (- x + 1\right )^{\frac{11}{2}} \left (x + 1\right )^{\frac{7}{2}}}{9} + \frac{11 \left (- x + 1\right )^{\frac{9}{2}} \left (x + 1\right )^{\frac{7}{2}}}{72} + \frac{11 \left (- x + 1\right )^{\frac{7}{2}} \left (x + 1\right )^{\frac{7}{2}}}{56} + \frac{55 \operatorname{asin}{\left (x \right )}}{128} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-x)**(11/2)*(1+x)**(5/2),x)

[Out]

11*x*(-x + 1)**(5/2)*(x + 1)**(5/2)/48 + 55*x*(-x + 1)**(3/2)*(x + 1)**(3/2)/192
+ 55*x*sqrt(-x + 1)*sqrt(x + 1)/128 + (-x + 1)**(11/2)*(x + 1)**(7/2)/9 + 11*(-
x + 1)**(9/2)*(x + 1)**(7/2)/72 + 11*(-x + 1)**(7/2)*(x + 1)**(7/2)/56 + 55*asin
(x)/128

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Mathematica [A]  time = 0.0610131, size = 74, normalized size = 0.57 \[ \frac{\sqrt{1-x^2} \left (-896 x^8+3024 x^7-1024 x^6-7224 x^5+8448 x^4+3066 x^3-10240 x^2+4599 x+3712\right )}{8064}+\frac{55}{64} \sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - x)^(11/2)*(1 + x)^(5/2),x]

[Out]

(Sqrt[1 - x^2]*(3712 + 4599*x - 10240*x^2 + 3066*x^3 + 8448*x^4 - 7224*x^5 - 102
4*x^6 + 3024*x^7 - 896*x^8))/8064 + (55*ArcSin[Sqrt[1 + x]/Sqrt[2]])/64

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Maple [A]  time = 0.007, size = 155, normalized size = 1.2 \[{\frac{1}{9} \left ( 1-x \right ) ^{{\frac{11}{2}}} \left ( 1+x \right ) ^{{\frac{7}{2}}}}+{\frac{11}{72} \left ( 1-x \right ) ^{{\frac{9}{2}}} \left ( 1+x \right ) ^{{\frac{7}{2}}}}+{\frac{11}{56} \left ( 1-x \right ) ^{{\frac{7}{2}}} \left ( 1+x \right ) ^{{\frac{7}{2}}}}+{\frac{11}{48} \left ( 1-x \right ) ^{{\frac{5}{2}}} \left ( 1+x \right ) ^{{\frac{7}{2}}}}+{\frac{11}{48} \left ( 1-x \right ) ^{{\frac{3}{2}}} \left ( 1+x \right ) ^{{\frac{7}{2}}}}+{\frac{11}{64}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{7}{2}}}}-{\frac{11}{192}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{5}{2}}}}-{\frac{55}{384}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{3}{2}}}}-{\frac{55}{128}\sqrt{1-x}\sqrt{1+x}}+{\frac{55\,\arcsin \left ( x \right ) }{128}\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-x)^(11/2)*(1+x)^(5/2),x)

[Out]

1/9*(1-x)^(11/2)*(1+x)^(7/2)+11/72*(1-x)^(9/2)*(1+x)^(7/2)+11/56*(1-x)^(7/2)*(1+
x)^(7/2)+11/48*(1-x)^(5/2)*(1+x)^(7/2)+11/48*(1-x)^(3/2)*(1+x)^(7/2)+11/64*(1-x)
^(1/2)*(1+x)^(7/2)-11/192*(1-x)^(1/2)*(1+x)^(5/2)-55/384*(1-x)^(1/2)*(1+x)^(3/2)
-55/128*(1-x)^(1/2)*(1+x)^(1/2)+55/128*((1+x)*(1-x))^(1/2)/(1+x)^(1/2)/(1-x)^(1/
2)*arcsin(x)

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Maxima [A]  time = 1.50122, size = 105, normalized size = 0.81 \[ \frac{1}{9} \,{\left (-x^{2} + 1\right )}^{\frac{7}{2}} x^{2} - \frac{3}{8} \,{\left (-x^{2} + 1\right )}^{\frac{7}{2}} x + \frac{29}{63} \,{\left (-x^{2} + 1\right )}^{\frac{7}{2}} + \frac{11}{48} \,{\left (-x^{2} + 1\right )}^{\frac{5}{2}} x + \frac{55}{192} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} x + \frac{55}{128} \, \sqrt{-x^{2} + 1} x + \frac{55}{128} \, \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^(5/2)*(-x + 1)^(11/2),x, algorithm="maxima")

[Out]

1/9*(-x^2 + 1)^(7/2)*x^2 - 3/8*(-x^2 + 1)^(7/2)*x + 29/63*(-x^2 + 1)^(7/2) + 11/
48*(-x^2 + 1)^(5/2)*x + 55/192*(-x^2 + 1)^(3/2)*x + 55/128*sqrt(-x^2 + 1)*x + 55
/128*arcsin(x)

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Fricas [A]  time = 0.216872, size = 432, normalized size = 3.32 \[ -\frac{896 \, x^{18} - 3024 \, x^{17} - 35712 \, x^{16} + 131208 \, x^{15} + 200448 \, x^{14} - 1145970 \, x^{13} + 26880 \, x^{12} + 4224339 \, x^{11} - 2862720 \, x^{10} - 7768929 \, x^{9} + 9289728 \, x^{8} + 6681528 \, x^{7} - 13848576 \, x^{6} - 843696 \, x^{5} + 10321920 \, x^{4} - 2452800 \, x^{3} - 3096576 \, x^{2} + 3 \,{\left (2688 \, x^{16} - 9072 \, x^{15} - 32768 \, x^{14} + 142632 \, x^{13} + 62720 \, x^{12} - 733614 \, x^{11} + 344064 \, x^{10} + 1729707 \, x^{9} - 1756160 \, x^{8} - 1902600 \, x^{7} + 3282944 \, x^{6} + 542864 \, x^{5} - 2924544 \, x^{4} + 621376 \, x^{3} + 1032192 \, x^{2} - 392448 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} + 6930 \,{\left (9 \, x^{8} - 120 \, x^{6} + 432 \, x^{4} - 576 \, x^{2} -{\left (x^{8} - 40 \, x^{6} + 240 \, x^{4} - 448 \, x^{2} + 256\right )} \sqrt{x + 1} \sqrt{-x + 1} + 256\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) + 1177344 \, x}{8064 \,{\left (9 \, x^{8} - 120 \, x^{6} + 432 \, x^{4} - 576 \, x^{2} -{\left (x^{8} - 40 \, x^{6} + 240 \, x^{4} - 448 \, x^{2} + 256\right )} \sqrt{x + 1} \sqrt{-x + 1} + 256\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^(5/2)*(-x + 1)^(11/2),x, algorithm="fricas")

[Out]

-1/8064*(896*x^18 - 3024*x^17 - 35712*x^16 + 131208*x^15 + 200448*x^14 - 1145970
*x^13 + 26880*x^12 + 4224339*x^11 - 2862720*x^10 - 7768929*x^9 + 9289728*x^8 + 6
681528*x^7 - 13848576*x^6 - 843696*x^5 + 10321920*x^4 - 2452800*x^3 - 3096576*x^
2 + 3*(2688*x^16 - 9072*x^15 - 32768*x^14 + 142632*x^13 + 62720*x^12 - 733614*x^
11 + 344064*x^10 + 1729707*x^9 - 1756160*x^8 - 1902600*x^7 + 3282944*x^6 + 54286
4*x^5 - 2924544*x^4 + 621376*x^3 + 1032192*x^2 - 392448*x)*sqrt(x + 1)*sqrt(-x +
1) + 6930*(9*x^8 - 120*x^6 + 432*x^4 - 576*x^2 - (x^8 - 40*x^6 + 240*x^4 - 448*
x^2 + 256)*sqrt(x + 1)*sqrt(-x + 1) + 256)*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)
/x) + 1177344*x)/(9*x^8 - 120*x^6 + 432*x^4 - 576*x^2 - (x^8 - 40*x^6 + 240*x^4
- 448*x^2 + 256)*sqrt(x + 1)*sqrt(-x + 1) + 256)

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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)**(11/2)*(1+x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.257045, size = 409, normalized size = 3.15 \[ -\frac{1}{315} \,{\left ({\left ({\left ({\left (5 \,{\left ({\left (7 \,{\left (x + 1\right )}{\left (x - 7\right )} + 195\right )}{\left (x + 1\right )} - 386\right )}{\left (x + 1\right )} + 2369\right )}{\left (x + 1\right )} - 1836\right )}{\left (x + 1\right )} + 861\right )}{\left (x + 1\right )} - 210\right )}{\left (x + 1\right )}^{\frac{3}{2}} \sqrt{-x + 1} - \frac{1}{105} \,{\left ({\left (3 \,{\left ({\left (5 \,{\left (x + 1\right )}{\left (x - 5\right )} + 74\right )}{\left (x + 1\right )} - 96\right )}{\left (x + 1\right )} + 203\right )}{\left (x + 1\right )} - 70\right )}{\left (x + 1\right )}^{\frac{3}{2}} \sqrt{-x + 1} + \frac{1}{3} \,{\left ({\left (3 \,{\left (x + 1\right )}{\left (x - 3\right )} + 17\right )}{\left (x + 1\right )} - 10\right )}{\left (x + 1\right )}^{\frac{3}{2}} \sqrt{-x + 1} -{\left (x + 1\right )}^{\frac{3}{2}}{\left (x - 1\right )} \sqrt{-x + 1} + \frac{1}{128} \,{\left ({\left (2 \,{\left ({\left (4 \,{\left ({\left (6 \,{\left (x + 1\right )}{\left (x - 6\right )} + 125\right )}{\left (x + 1\right )} - 205\right )}{\left (x + 1\right )} + 795\right )}{\left (x + 1\right )} - 449\right )}{\left (x + 1\right )} + 251\right )}{\left (x + 1\right )} - 15\right )} \sqrt{x + 1} \sqrt{-x + 1} - \frac{5}{48} \,{\left ({\left (2 \,{\left ({\left (4 \,{\left (x + 1\right )}{\left (x - 4\right )} + 39\right )}{\left (x + 1\right )} - 37\right )}{\left (x + 1\right )} + 31\right )}{\left (x + 1\right )} - 3\right )} \sqrt{x + 1} \sqrt{-x + 1} + \frac{1}{8} \,{\left ({\left (2 \,{\left (x + 1\right )}{\left (x - 2\right )} + 5\right )}{\left (x + 1\right )} - 1\right )} \sqrt{x + 1} \sqrt{-x + 1} + \frac{1}{2} \, \sqrt{x + 1} x \sqrt{-x + 1} + \frac{55}{64} \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^(5/2)*(-x + 1)^(11/2),x, algorithm="giac")

[Out]

-1/315*((((5*((7*(x + 1)*(x - 7) + 195)*(x + 1) - 386)*(x + 1) + 2369)*(x + 1) -
1836)*(x + 1) + 861)*(x + 1) - 210)*(x + 1)^(3/2)*sqrt(-x + 1) - 1/105*((3*((5*
(x + 1)*(x - 5) + 74)*(x + 1) - 96)*(x + 1) + 203)*(x + 1) - 70)*(x + 1)^(3/2)*s
qrt(-x + 1) + 1/3*((3*(x + 1)*(x - 3) + 17)*(x + 1) - 10)*(x + 1)^(3/2)*sqrt(-x
+ 1) - (x + 1)^(3/2)*(x - 1)*sqrt(-x + 1) + 1/128*((2*((4*((6*(x + 1)*(x - 6) +
125)*(x + 1) - 205)*(x + 1) + 795)*(x + 1) - 449)*(x + 1) + 251)*(x + 1) - 15)*s
qrt(x + 1)*sqrt(-x + 1) - 5/48*((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) + 55/64*arc
sin(1/2*sqrt(2)*sqrt(x + 1))