3.1103 \(\int \frac{(1+x)^{5/2}}{(1-x)^{19/2}} \, dx\)

Optimal. Leaf size=121 \[ \frac{8 (x+1)^{7/2}}{153153 (1-x)^{7/2}}+\frac{8 (x+1)^{7/2}}{21879 (1-x)^{9/2}}+\frac{4 (x+1)^{7/2}}{2431 (1-x)^{11/2}}+\frac{4 (x+1)^{7/2}}{663 (1-x)^{13/2}}+\frac{(x+1)^{7/2}}{51 (1-x)^{15/2}}+\frac{(x+1)^{7/2}}{17 (1-x)^{17/2}} \]

[Out]

(1 + x)^(7/2)/(17*(1 - x)^(17/2)) + (1 + x)^(7/2)/(51*(1 - x)^(15/2)) + (4*(1 +
x)^(7/2))/(663*(1 - x)^(13/2)) + (4*(1 + x)^(7/2))/(2431*(1 - x)^(11/2)) + (8*(1
+ x)^(7/2))/(21879*(1 - x)^(9/2)) + (8*(1 + x)^(7/2))/(153153*(1 - x)^(7/2))

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Rubi [A]  time = 0.0880289, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{8 (x+1)^{7/2}}{153153 (1-x)^{7/2}}+\frac{8 (x+1)^{7/2}}{21879 (1-x)^{9/2}}+\frac{4 (x+1)^{7/2}}{2431 (1-x)^{11/2}}+\frac{4 (x+1)^{7/2}}{663 (1-x)^{13/2}}+\frac{(x+1)^{7/2}}{51 (1-x)^{15/2}}+\frac{(x+1)^{7/2}}{17 (1-x)^{17/2}} \]

Antiderivative was successfully verified.

[In]  Int[(1 + x)^(5/2)/(1 - x)^(19/2),x]

[Out]

(1 + x)^(7/2)/(17*(1 - x)^(17/2)) + (1 + x)^(7/2)/(51*(1 - x)^(15/2)) + (4*(1 +
x)^(7/2))/(663*(1 - x)^(13/2)) + (4*(1 + x)^(7/2))/(2431*(1 - x)^(11/2)) + (8*(1
+ x)^(7/2))/(21879*(1 - x)^(9/2)) + (8*(1 + x)^(7/2))/(153153*(1 - x)^(7/2))

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Rubi in Sympy [A]  time = 10.4092, size = 97, normalized size = 0.8 \[ \frac{8 \left (x + 1\right )^{\frac{7}{2}}}{153153 \left (- x + 1\right )^{\frac{7}{2}}} + \frac{8 \left (x + 1\right )^{\frac{7}{2}}}{21879 \left (- x + 1\right )^{\frac{9}{2}}} + \frac{4 \left (x + 1\right )^{\frac{7}{2}}}{2431 \left (- x + 1\right )^{\frac{11}{2}}} + \frac{4 \left (x + 1\right )^{\frac{7}{2}}}{663 \left (- x + 1\right )^{\frac{13}{2}}} + \frac{\left (x + 1\right )^{\frac{7}{2}}}{51 \left (- x + 1\right )^{\frac{15}{2}}} + \frac{\left (x + 1\right )^{\frac{7}{2}}}{17 \left (- x + 1\right )^{\frac{17}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+x)**(5/2)/(1-x)**(19/2),x)

[Out]

8*(x + 1)**(7/2)/(153153*(-x + 1)**(7/2)) + 8*(x + 1)**(7/2)/(21879*(-x + 1)**(9
/2)) + 4*(x + 1)**(7/2)/(2431*(-x + 1)**(11/2)) + 4*(x + 1)**(7/2)/(663*(-x + 1)
**(13/2)) + (x + 1)**(7/2)/(51*(-x + 1)**(15/2)) + (x + 1)**(7/2)/(17*(-x + 1)**
(17/2))

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Mathematica [A]  time = 0.0287652, size = 50, normalized size = 0.41 \[ \frac{(x+1)^3 \sqrt{1-x^2} \left (8 x^5-96 x^4+556 x^3-2096 x^2+5871 x-13252\right )}{153153 (x-1)^9} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(1 + x)^(5/2)/(1 - x)^(19/2),x]

[Out]

((1 + x)^3*Sqrt[1 - x^2]*(-13252 + 5871*x - 2096*x^2 + 556*x^3 - 96*x^4 + 8*x^5)
)/(153153*(-1 + x)^9)

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Maple [A]  time = 0.006, size = 40, normalized size = 0.3 \[ -{\frac{8\,{x}^{5}-96\,{x}^{4}+556\,{x}^{3}-2096\,{x}^{2}+5871\,x-13252}{153153} \left ( 1+x \right ) ^{{\frac{7}{2}}} \left ( 1-x \right ) ^{-{\frac{17}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x)^(5/2)/(1-x)^(19/2),x)

[Out]

-1/153153*(1+x)^(7/2)*(8*x^5-96*x^4+556*x^3-2096*x^2+5871*x-13252)/(1-x)^(17/2)

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Maxima [A]  time = 1.39437, size = 610, normalized size = 5.04 \[ -\frac{{\left (-x^{2} + 1\right )}^{\frac{5}{2}}}{6 \,{\left (x^{11} - 11 \, x^{10} + 55 \, x^{9} - 165 \, x^{8} + 330 \, x^{7} - 462 \, x^{6} + 462 \, x^{5} - 330 \, x^{4} + 165 \, x^{3} - 55 \, x^{2} + 11 \, x - 1\right )}} - \frac{5 \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}}}{42 \,{\left (x^{10} - 10 \, x^{9} + 45 \, x^{8} - 120 \, x^{7} + 210 \, x^{6} - 252 \, x^{5} + 210 \, x^{4} - 120 \, x^{3} + 45 \, x^{2} - 10 \, x + 1\right )}} - \frac{5 \, \sqrt{-x^{2} + 1}}{119 \,{\left (x^{9} - 9 \, x^{8} + 36 \, x^{7} - 84 \, x^{6} + 126 \, x^{5} - 126 \, x^{4} + 84 \, x^{3} - 36 \, x^{2} + 9 \, x - 1\right )}} - \frac{\sqrt{-x^{2} + 1}}{714 \,{\left (x^{8} - 8 \, x^{7} + 28 \, x^{6} - 56 \, x^{5} + 70 \, x^{4} - 56 \, x^{3} + 28 \, x^{2} - 8 \, x + 1\right )}} + \frac{\sqrt{-x^{2} + 1}}{1326 \,{\left (x^{7} - 7 \, x^{6} + 21 \, x^{5} - 35 \, x^{4} + 35 \, x^{3} - 21 \, x^{2} + 7 \, x - 1\right )}} - \frac{\sqrt{-x^{2} + 1}}{2431 \,{\left (x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1\right )}} + \frac{5 \, \sqrt{-x^{2} + 1}}{21879 \,{\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} - \frac{20 \, \sqrt{-x^{2} + 1}}{153153 \,{\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} + \frac{4 \, \sqrt{-x^{2} + 1}}{51051 \,{\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} - \frac{8 \, \sqrt{-x^{2} + 1}}{153153 \,{\left (x^{2} - 2 \, x + 1\right )}} + \frac{8 \, \sqrt{-x^{2} + 1}}{153153 \,{\left (x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^(5/2)/(-x + 1)^(19/2),x, algorithm="maxima")

[Out]

-1/6*(-x^2 + 1)^(5/2)/(x^11 - 11*x^10 + 55*x^9 - 165*x^8 + 330*x^7 - 462*x^6 + 4
62*x^5 - 330*x^4 + 165*x^3 - 55*x^2 + 11*x - 1) - 5/42*(-x^2 + 1)^(3/2)/(x^10 -
10*x^9 + 45*x^8 - 120*x^7 + 210*x^6 - 252*x^5 + 210*x^4 - 120*x^3 + 45*x^2 - 10*
x + 1) - 5/119*sqrt(-x^2 + 1)/(x^9 - 9*x^8 + 36*x^7 - 84*x^6 + 126*x^5 - 126*x^4
+ 84*x^3 - 36*x^2 + 9*x - 1) - 1/714*sqrt(-x^2 + 1)/(x^8 - 8*x^7 + 28*x^6 - 56*
x^5 + 70*x^4 - 56*x^3 + 28*x^2 - 8*x + 1) + 1/1326*sqrt(-x^2 + 1)/(x^7 - 7*x^6 +
21*x^5 - 35*x^4 + 35*x^3 - 21*x^2 + 7*x - 1) - 1/2431*sqrt(-x^2 + 1)/(x^6 - 6*x
^5 + 15*x^4 - 20*x^3 + 15*x^2 - 6*x + 1) + 5/21879*sqrt(-x^2 + 1)/(x^5 - 5*x^4 +
10*x^3 - 10*x^2 + 5*x - 1) - 20/153153*sqrt(-x^2 + 1)/(x^4 - 4*x^3 + 6*x^2 - 4*
x + 1) + 4/51051*sqrt(-x^2 + 1)/(x^3 - 3*x^2 + 3*x - 1) - 8/153153*sqrt(-x^2 + 1
)/(x^2 - 2*x + 1) + 8/153153*sqrt(-x^2 + 1)/(x - 1)

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Fricas [A]  time = 0.212167, size = 473, normalized size = 3.91 \[ \frac{13244 \, x^{17} - 225284 \, x^{16} + 901748 \, x^{15} + 897872 \, x^{14} - 14864103 \, x^{13} + 36507432 \, x^{12} - 9937486 \, x^{11} - 112321924 \, x^{10} + 214227013 \, x^{9} - 89091288 \, x^{8} - 223632552 \, x^{7} + 373284912 \, x^{6} - 156011856 \, x^{5} - 114354240 \, x^{4} + 153561408 \, x^{3} - 98017920 \, x^{2} + 17 \,{\left (780 \, x^{16} - 8 \, x^{15} - 59212 \, x^{14} + 318076 \, x^{13} - 482261 \, x^{12} - 952835 \, x^{11} + 4671953 \, x^{10} - 6036173 \, x^{9} - 1413984 \, x^{8} + 13635336 \, x^{7} - 16432416 \, x^{6} + 3795792 \, x^{5} + 9609600 \, x^{4} - 10186176 \, x^{3} + 5765760 \, x^{2} - 2306304 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} + 39207168 \, x}{153153 \,{\left (x^{17} - 17 \, x^{16} + 68 \, x^{15} + 68 \, x^{14} - 1122 \, x^{13} + 2754 \, x^{12} - 748 \, x^{11} - 8500 \, x^{10} + 16201 \, x^{9} - 6409 \, x^{8} - 16864 \, x^{7} + 27064 \, x^{6} - 12512 \, x^{5} - 7344 \, x^{4} + 13056 \, x^{3} - 7616 \, x^{2} +{\left (x^{16} - 76 \, x^{14} + 408 \, x^{13} - 618 \, x^{12} - 1224 \, x^{11} + 5996 \, x^{10} - 7752 \, x^{9} - 1919 \, x^{8} + 17544 \, x^{7} - 20456 \, x^{6} + 5168 \, x^{5} + 11248 \, x^{4} - 14144 \, x^{3} + 7744 \, x^{2} - 2176 \, x + 256\right )} \sqrt{x + 1} \sqrt{-x + 1} + 2176 \, x - 256\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^(5/2)/(-x + 1)^(19/2),x, algorithm="fricas")

[Out]

1/153153*(13244*x^17 - 225284*x^16 + 901748*x^15 + 897872*x^14 - 14864103*x^13 +
36507432*x^12 - 9937486*x^11 - 112321924*x^10 + 214227013*x^9 - 89091288*x^8 -
223632552*x^7 + 373284912*x^6 - 156011856*x^5 - 114354240*x^4 + 153561408*x^3 -
98017920*x^2 + 17*(780*x^16 - 8*x^15 - 59212*x^14 + 318076*x^13 - 482261*x^12 -
952835*x^11 + 4671953*x^10 - 6036173*x^9 - 1413984*x^8 + 13635336*x^7 - 16432416
*x^6 + 3795792*x^5 + 9609600*x^4 - 10186176*x^3 + 5765760*x^2 - 2306304*x)*sqrt(
x + 1)*sqrt(-x + 1) + 39207168*x)/(x^17 - 17*x^16 + 68*x^15 + 68*x^14 - 1122*x^1
3 + 2754*x^12 - 748*x^11 - 8500*x^10 + 16201*x^9 - 6409*x^8 - 16864*x^7 + 27064*
x^6 - 12512*x^5 - 7344*x^4 + 13056*x^3 - 7616*x^2 + (x^16 - 76*x^14 + 408*x^13 -
618*x^12 - 1224*x^11 + 5996*x^10 - 7752*x^9 - 1919*x^8 + 17544*x^7 - 20456*x^6
+ 5168*x^5 + 11248*x^4 - 14144*x^3 + 7744*x^2 - 2176*x + 256)*sqrt(x + 1)*sqrt(-
x + 1) + 2176*x - 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)**(5/2)/(1-x)**(19/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.230499, size = 65, normalized size = 0.54 \[ \frac{{\left ({\left (4 \,{\left ({\left (2 \,{\left (x + 1\right )}{\left (x - 16\right )} + 255\right )}{\left (x + 1\right )} - 1105\right )}{\left (x + 1\right )} + 12155\right )}{\left (x + 1\right )} - 21879\right )}{\left (x + 1\right )}^{\frac{7}{2}} \sqrt{-x + 1}}{153153 \,{\left (x - 1\right )}^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^(5/2)/(-x + 1)^(19/2),x, algorithm="giac")

[Out]

1/153153*((4*((2*(x + 1)*(x - 16) + 255)*(x + 1) - 1105)*(x + 1) + 12155)*(x + 1
) - 21879)*(x + 1)^(7/2)*sqrt(-x + 1)/(x - 1)^9