3.1115 \(\int \frac{1}{(1-x)^{11/2} \sqrt{1+x}} \, dx\)

Optimal. Leaf size=101 \[ \frac{8 \sqrt{x+1}}{315 \sqrt{1-x}}+\frac{8 \sqrt{x+1}}{315 (1-x)^{3/2}}+\frac{4 \sqrt{x+1}}{105 (1-x)^{5/2}}+\frac{4 \sqrt{x+1}}{63 (1-x)^{7/2}}+\frac{\sqrt{x+1}}{9 (1-x)^{9/2}} \]

[Out]

Sqrt[1 + x]/(9*(1 - x)^(9/2)) + (4*Sqrt[1 + x])/(63*(1 - x)^(7/2)) + (4*Sqrt[1 +
x])/(105*(1 - x)^(5/2)) + (8*Sqrt[1 + x])/(315*(1 - x)^(3/2)) + (8*Sqrt[1 + x])
/(315*Sqrt[1 - x])

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Rubi [A]  time = 0.0673135, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{8 \sqrt{x+1}}{315 \sqrt{1-x}}+\frac{8 \sqrt{x+1}}{315 (1-x)^{3/2}}+\frac{4 \sqrt{x+1}}{105 (1-x)^{5/2}}+\frac{4 \sqrt{x+1}}{63 (1-x)^{7/2}}+\frac{\sqrt{x+1}}{9 (1-x)^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[1 + x]/(9*(1 - x)^(9/2)) + (4*Sqrt[1 + x])/(63*(1 - x)^(7/2)) + (4*Sqrt[1 +
x])/(105*(1 - x)^(5/2)) + (8*Sqrt[1 + x])/(315*(1 - x)^(3/2)) + (8*Sqrt[1 + x])
/(315*Sqrt[1 - x])

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Rubi in Sympy [A]  time = 7.8595, size = 82, normalized size = 0.81 \[ \frac{8 \sqrt{x + 1}}{315 \sqrt{- x + 1}} + \frac{8 \sqrt{x + 1}}{315 \left (- x + 1\right )^{\frac{3}{2}}} + \frac{4 \sqrt{x + 1}}{105 \left (- x + 1\right )^{\frac{5}{2}}} + \frac{4 \sqrt{x + 1}}{63 \left (- x + 1\right )^{\frac{7}{2}}} + \frac{\sqrt{x + 1}}{9 \left (- x + 1\right )^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8*sqrt(x + 1)/(315*sqrt(-x + 1)) + 8*sqrt(x + 1)/(315*(-x + 1)**(3/2)) + 4*sqrt(
x + 1)/(105*(-x + 1)**(5/2)) + 4*sqrt(x + 1)/(63*(-x + 1)**(7/2)) + sqrt(x + 1)/
(9*(-x + 1)**(9/2))

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Mathematica [A]  time = 0.0250067, size = 40, normalized size = 0.4 \[ -\frac{\sqrt{1-x^2} \left (8 x^4-40 x^3+84 x^2-100 x+83\right )}{315 (x-1)^5} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - x^2]*(83 - 100*x + 84*x^2 - 40*x^3 + 8*x^4))/(315*(-1 + x)^5)

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Maple [A]  time = 0.006, size = 35, normalized size = 0.4 \[{\frac{8\,{x}^{4}-40\,{x}^{3}+84\,{x}^{2}-100\,x+83}{315}\sqrt{1+x} \left ( 1-x \right ) ^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/315*(1+x)^(1/2)*(8*x^4-40*x^3+84*x^2-100*x+83)/(1-x)^(9/2)

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Maxima [A]  time = 1.51844, size = 177, normalized size = 1.75 \[ -\frac{\sqrt{-x^{2} + 1}}{9 \,{\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} + \frac{4 \, \sqrt{-x^{2} + 1}}{63 \,{\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} - \frac{4 \, \sqrt{-x^{2} + 1}}{105 \,{\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} + \frac{8 \, \sqrt{-x^{2} + 1}}{315 \,{\left (x^{2} - 2 \, x + 1\right )}} - \frac{8 \, \sqrt{-x^{2} + 1}}{315 \,{\left (x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9*sqrt(-x^2 + 1)/(x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5*x - 1) + 4/63*sqrt(-x^2 +
1)/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1) - 4/105*sqrt(-x^2 + 1)/(x^3 - 3*x^2 + 3*x -
1) + 8/315*sqrt(-x^2 + 1)/(x^2 - 2*x + 1) - 8/315*sqrt(-x^2 + 1)/(x - 1)

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Fricas [A]  time = 0.20443, size = 257, normalized size = 2.54 \[ \frac{91 \, x^{9} - 747 \, x^{8} + 1314 \, x^{7} + 1974 \, x^{6} - 8442 \, x^{5} + 7560 \, x^{4} + 3360 \, x^{3} - 10080 \, x^{2} + 3 \,{\left (25 \, x^{8} + 24 \, x^{7} - 658 \, x^{6} + 1624 \, x^{5} - 840 \, x^{4} - 1960 \, x^{3} + 3360 \, x^{2} - 1680 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} + 5040 \, x}{315 \,{\left (x^{9} - 9 \, x^{8} + 18 \, x^{7} + 18 \, x^{6} - 99 \, x^{5} + 99 \, x^{4} + 24 \, x^{3} - 108 \, x^{2} +{\left (x^{8} - 22 \, x^{6} + 60 \, x^{5} - 39 \, x^{4} - 60 \, x^{3} + 116 \, x^{2} - 72 \, x + 16\right )} \sqrt{x + 1} \sqrt{-x + 1} + 72 \, x - 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/315*(91*x^9 - 747*x^8 + 1314*x^7 + 1974*x^6 - 8442*x^5 + 7560*x^4 + 3360*x^3 -
10080*x^2 + 3*(25*x^8 + 24*x^7 - 658*x^6 + 1624*x^5 - 840*x^4 - 1960*x^3 + 3360
*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
+ 116*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/(1-x)**(11/2)/(1+x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.212374, size = 57, normalized size = 0.56 \[ -\frac{{\left (4 \,{\left ({\left (2 \,{\left (x + 1\right )}{\left (x - 8\right )} + 63\right )}{\left (x + 1\right )} - 105\right )}{\left (x + 1\right )} + 315\right )} \sqrt{x + 1} \sqrt{-x + 1}}{315 \,{\left (x - 1\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/315*(4*((2*(x + 1)*(x - 8) + 63)*(x + 1) - 105)*(x + 1) + 315)*sqrt(x + 1)*sq
rt(-x + 1)/(x - 1)^5