3.1085 \(\int \frac{(1+x)^{3/2}}{(1-x)^{11/2}} \, dx\)

Optimal. Leaf size=61 \[ \frac{2 (x+1)^{5/2}}{315 (1-x)^{5/2}}+\frac{2 (x+1)^{5/2}}{63 (1-x)^{7/2}}+\frac{(x+1)^{5/2}}{9 (1-x)^{9/2}} \]

[Out]

(1 + x)^(5/2)/(9*(1 - x)^(9/2)) + (2*(1 + x)^(5/2))/(63*(1 - x)^(7/2)) + (2*(1 +
x)^(5/2))/(315*(1 - x)^(5/2))

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Rubi [A]  time = 0.0374626, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{2 (x+1)^{5/2}}{315 (1-x)^{5/2}}+\frac{2 (x+1)^{5/2}}{63 (1-x)^{7/2}}+\frac{(x+1)^{5/2}}{9 (1-x)^{9/2}} \]

Antiderivative was successfully verified.

[In]  Int[(1 + x)^(3/2)/(1 - x)^(11/2),x]

[Out]

(1 + x)^(5/2)/(9*(1 - x)^(9/2)) + (2*(1 + x)^(5/2))/(63*(1 - x)^(7/2)) + (2*(1 +
x)^(5/2))/(315*(1 - x)^(5/2))

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Rubi in Sympy [A]  time = 5.12022, size = 48, normalized size = 0.79 \[ \frac{2 \left (x + 1\right )^{\frac{5}{2}}}{315 \left (- x + 1\right )^{\frac{5}{2}}} + \frac{2 \left (x + 1\right )^{\frac{5}{2}}}{63 \left (- x + 1\right )^{\frac{7}{2}}} + \frac{\left (x + 1\right )^{\frac{5}{2}}}{9 \left (- x + 1\right )^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+x)**(3/2)/(1-x)**(11/2),x)

[Out]

2*(x + 1)**(5/2)/(315*(-x + 1)**(5/2)) + 2*(x + 1)**(5/2)/(63*(-x + 1)**(7/2)) +
(x + 1)**(5/2)/(9*(-x + 1)**(9/2))

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Mathematica [A]  time = 0.0214625, size = 35, normalized size = 0.57 \[ -\frac{(x+1)^2 \sqrt{1-x^2} \left (2 x^2-14 x+47\right )}{315 (x-1)^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + x)^(3/2)/(1 - x)^(11/2),x]

[Out]

-((1 + x)^2*Sqrt[1 - x^2]*(47 - 14*x + 2*x^2))/(315*(-1 + x)^5)

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Maple [A]  time = 0.004, size = 25, normalized size = 0.4 \[{\frac{2\,{x}^{2}-14\,x+47}{315} \left ( 1+x \right ) ^{{\frac{5}{2}}} \left ( 1-x \right ) ^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x)^(3/2)/(1-x)^(11/2),x)

[Out]

1/315*(1+x)^(5/2)*(2*x^2-14*x+47)/(1-x)^(9/2)

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Maxima [A]  time = 1.34682, size = 232, normalized size = 3.8 \[ \frac{{\left (-x^{2} + 1\right )}^{\frac{3}{2}}}{3 \,{\left (x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1\right )}} + \frac{2 \, \sqrt{-x^{2} + 1}}{9 \,{\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} + \frac{\sqrt{-x^{2} + 1}}{63 \,{\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} - \frac{\sqrt{-x^{2} + 1}}{105 \,{\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} + \frac{2 \, \sqrt{-x^{2} + 1}}{315 \,{\left (x^{2} - 2 \, x + 1\right )}} - \frac{2 \, \sqrt{-x^{2} + 1}}{315 \,{\left (x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^(3/2)/(-x + 1)^(11/2),x, algorithm="maxima")

[Out]

1/3*(-x^2 + 1)^(3/2)/(x^6 - 6*x^5 + 15*x^4 - 20*x^3 + 15*x^2 - 6*x + 1) + 2/9*sq
rt(-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 = 0.203186, size = 257, normalized size = 4.21 \[ \frac{49 \, x^{9} - 423 \, x^{8} + 801 \, x^{7} + 1071 \, x^{6} - 4158 \, x^{5} + 3780 \, x^{4} - 840 \, x^{3} - 5040 \, x^{2} + 3 \,{\left (15 \, x^{8} + 6 \, x^{7} - 357 \, x^{6} + 896 \, x^{5} - 420 \, x^{4} - 560 \, x^{3} + 1680 \, x^{2} - 1680 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} + 5040 \, x}{315 \,{\left (x^{9} - 9 \, x^{8} + 18 \, x^{7} + 18 \, x^{6} - 99 \, x^{5} + 99 \, x^{4} + 24 \, x^{3} - 108 \, x^{2} +{\left (x^{8} - 22 \, x^{6} + 60 \, x^{5} - 39 \, x^{4} - 60 \, x^{3} + 116 \, x^{2} - 72 \, x + 16\right )} \sqrt{x + 1} \sqrt{-x + 1} + 72 \, x - 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^(3/2)/(-x + 1)^(11/2),x, algorithm="fricas")

[Out]

1/315*(49*x^9 - 423*x^8 + 801*x^7 + 1071*x^6 - 4158*x^5 + 3780*x^4 - 840*x^3 - 5
040*x^2 + 3*(15*x^8 + 6*x^7 - 357*x^6 + 896*x^5 - 420*x^4 - 560*x^3 + 1680*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+x)**(3/2)/(1-x)**(11/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.215412, size = 39, normalized size = 0.64 \[ -\frac{{\left (2 \,{\left (x + 1\right )}{\left (x - 8\right )} + 63\right )}{\left (x + 1\right )}^{\frac{5}{2}} \sqrt{-x + 1}}{315 \,{\left (x - 1\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^(3/2)/(-x + 1)^(11/2),x, algorithm="giac")

[Out]

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