3.1073 \(\int \frac{\sqrt{1+x}}{(1-x)^{11/2}} \, dx\)

Optimal. Leaf size=81 \[ \frac{2 (x+1)^{3/2}}{315 (1-x)^{3/2}}+\frac{2 (x+1)^{3/2}}{105 (1-x)^{5/2}}+\frac{(x+1)^{3/2}}{21 (1-x)^{7/2}}+\frac{(x+1)^{3/2}}{9 (1-x)^{9/2}} \]

[Out]

(1 + x)^(3/2)/(9*(1 - x)^(9/2)) + (1 + x)^(3/2)/(21*(1 - x)^(7/2)) + (2*(1 + x)^
(3/2))/(105*(1 - x)^(5/2)) + (2*(1 + x)^(3/2))/(315*(1 - x)^(3/2))

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Rubi [A]  time = 0.0513608, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{2 (x+1)^{3/2}}{315 (1-x)^{3/2}}+\frac{2 (x+1)^{3/2}}{105 (1-x)^{5/2}}+\frac{(x+1)^{3/2}}{21 (1-x)^{7/2}}+\frac{(x+1)^{3/2}}{9 (1-x)^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(1 + x)^(3/2)/(9*(1 - x)^(9/2)) + (1 + x)^(3/2)/(21*(1 - x)^(7/2)) + (2*(1 + x)^
(3/2))/(105*(1 - x)^(5/2)) + (2*(1 + x)^(3/2))/(315*(1 - x)^(3/2))

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Rubi in Sympy [A]  time = 7.45147, size = 63, normalized size = 0.78 \[ \frac{2 \left (x + 1\right )^{\frac{3}{2}}}{315 \left (- x + 1\right )^{\frac{3}{2}}} + \frac{2 \left (x + 1\right )^{\frac{3}{2}}}{105 \left (- x + 1\right )^{\frac{5}{2}}} + \frac{\left (x + 1\right )^{\frac{3}{2}}}{21 \left (- x + 1\right )^{\frac{7}{2}}} + \frac{\left (x + 1\right )^{\frac{3}{2}}}{9 \left (- x + 1\right )^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(x + 1)**(3/2)/(315*(-x + 1)**(3/2)) + 2*(x + 1)**(3/2)/(105*(-x + 1)**(5/2))
+ (x + 1)**(3/2)/(21*(-x + 1)**(7/2)) + (x + 1)**(3/2)/(9*(-x + 1)**(9/2))

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Mathematica [A]  time = 0.0202197, size = 40, normalized size = 0.49 \[ \frac{\sqrt{1-x^2} \left (2 x^4-10 x^3+21 x^2-25 x-58\right )}{315 (x-1)^5} \]

Warning: Unable to verify antiderivative.

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

[Out]

(Sqrt[1 - x^2]*(-58 - 25*x + 21*x^2 - 10*x^3 + 2*x^4))/(315*(-1 + x)^5)

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Maple [A]  time = 0.005, size = 30, normalized size = 0.4 \[ -{\frac{2\,{x}^{3}-12\,{x}^{2}+33\,x-58}{315} \left ( 1+x \right ) ^{{\frac{3}{2}}} \left ( 1-x \right ) ^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x)^(1/2)/(1-x)^(11/2),x)

[Out]

-1/315*(1+x)^(3/2)*(2*x^3-12*x^2+33*x-58)/(1-x)^(9/2)

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Maxima [A]  time = 1.33516, size = 177, normalized size = 2.19 \[ -\frac{2 \, \sqrt{-x^{2} + 1}}{9 \,{\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} - \frac{\sqrt{-x^{2} + 1}}{63 \,{\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} + \frac{\sqrt{-x^{2} + 1}}{105 \,{\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} - \frac{2 \, \sqrt{-x^{2} + 1}}{315 \,{\left (x^{2} - 2 \, x + 1\right )}} + \frac{2 \, \sqrt{-x^{2} + 1}}{315 \,{\left (x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/9*sqrt(-x^2 + 1)/(x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5*x - 1) - 1/63*sqrt(-x^2 +
1)/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1) + 1/105*sqrt(-x^2 + 1)/(x^3 - 3*x^2 + 3*x -
1) - 2/315*sqrt(-x^2 + 1)/(x^2 - 2*x + 1) + 2/315*sqrt(-x^2 + 1)/(x - 1)

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Fricas [A]  time = 0.208992, size = 257, normalized size = 3.17 \[ \frac{56 \, x^{9} - 522 \, x^{8} + 1089 \, x^{7} + 924 \, x^{6} - 5607 \, x^{5} + 6300 \, x^{4} + 420 \, x^{3} - 7560 \, x^{2} + 3 \,{\left (20 \, x^{8} - 6 \, x^{7} - 413 \, x^{6} + 1169 \, x^{5} - 840 \, x^{4} - 980 \, x^{3} + 2520 \, x^{2} - 1680 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} + 5040 \, x}{315 \,{\left (x^{9} - 9 \, x^{8} + 18 \, x^{7} + 18 \, x^{6} - 99 \, x^{5} + 99 \, x^{4} + 24 \, x^{3} - 108 \, x^{2} +{\left (x^{8} - 22 \, x^{6} + 60 \, x^{5} - 39 \, x^{4} - 60 \, x^{3} + 116 \, x^{2} - 72 \, x + 16\right )} \sqrt{x + 1} \sqrt{-x + 1} + 72 \, x - 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/315*(56*x^9 - 522*x^8 + 1089*x^7 + 924*x^6 - 5607*x^5 + 6300*x^4 + 420*x^3 - 7
560*x^2 + 3*(20*x^8 - 6*x^7 - 413*x^6 + 1169*x^5 - 840*x^4 - 980*x^3 + 2520*x^2
- 1680*x)*sqrt(x + 1)*sqrt(-x + 1) + 5040*x)/(x^9 - 9*x^8 + 18*x^7 + 18*x^6 - 99
*x^5 + 99*x^4 + 24*x^3 - 108*x^2 + (x^8 - 22*x^6 + 60*x^5 - 39*x^4 - 60*x^3 + 11
6*x^2 - 72*x + 16)*sqrt(x + 1)*sqrt(-x + 1) + 72*x - 16)

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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)**(1/2)/(1-x)**(11/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.21288, size = 47, normalized size = 0.58 \[ \frac{{\left ({\left (2 \,{\left (x + 1\right )}{\left (x - 8\right )} + 63\right )}{\left (x + 1\right )} - 105\right )}{\left (x + 1\right )}^{\frac{3}{2}} \sqrt{-x + 1}}{315 \,{\left (x - 1\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/315*((2*(x + 1)*(x - 8) + 63)*(x + 1) - 105)*(x + 1)^(3/2)*sqrt(-x + 1)/(x - 1
)^5