3.1099 \(\int \frac{(1+x)^{5/2}}{(1-x)^{11/2}} \, dx\)

Optimal. Leaf size=41 \[ \frac{(x+1)^{7/2}}{63 (1-x)^{7/2}}+\frac{(x+1)^{7/2}}{9 (1-x)^{9/2}} \]

[Out]

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

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Rubi [A]  time = 0.0242483, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{(x+1)^{7/2}}{63 (1-x)^{7/2}}+\frac{(x+1)^{7/2}}{9 (1-x)^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 3.65636, size = 29, normalized size = 0.71 \[ \frac{\left (x + 1\right )^{\frac{7}{2}}}{63 \left (- x + 1\right )^{\frac{7}{2}}} + \frac{\left (x + 1\right )^{\frac{7}{2}}}{9 \left (- x + 1\right )^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0209838, size = 28, normalized size = 0.68 \[ \frac{(x-8) (x+1)^3 \sqrt{1-x^2}}{63 (x-1)^5} \]

Warning: Unable to verify antiderivative.

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

[Out]

((-8 + x)*(1 + x)^3*Sqrt[1 - x^2])/(63*(-1 + x)^5)

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Maple [A]  time = 0.003, size = 18, normalized size = 0.4 \[ -{\frac{x-8}{63} \left ( 1+x \right ) ^{{\frac{7}{2}}} \left ( 1-x \right ) ^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x)^(5/2)/(1-x)^(11/2),x)

[Out]

-1/63*(1+x)^(7/2)*(x-8)/(1-x)^(9/2)

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Maxima [A]  time = 1.35548, size = 294, normalized size = 7.17 \[ -\frac{{\left (-x^{2} + 1\right )}^{\frac{5}{2}}}{2 \,{\left (x^{7} - 7 \, x^{6} + 21 \, x^{5} - 35 \, x^{4} + 35 \, x^{3} - 21 \, x^{2} + 7 \, x - 1\right )}} - \frac{5 \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}}}{6 \,{\left (x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1\right )}} - \frac{5 \, \sqrt{-x^{2} + 1}}{9 \,{\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} - \frac{5 \, \sqrt{-x^{2} + 1}}{126 \,{\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} + \frac{\sqrt{-x^{2} + 1}}{42 \,{\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} - \frac{\sqrt{-x^{2} + 1}}{63 \,{\left (x^{2} - 2 \, x + 1\right )}} + \frac{\sqrt{-x^{2} + 1}}{63 \,{\left (x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(-x^2 + 1)^(5/2)/(x^7 - 7*x^6 + 21*x^5 - 35*x^4 + 35*x^3 - 21*x^2 + 7*x - 1
) - 5/6*(-x^2 + 1)^(3/2)/(x^6 - 6*x^5 + 15*x^4 - 20*x^3 + 15*x^2 - 6*x + 1) - 5/
9*sqrt(-x^2 + 1)/(x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5*x - 1) - 5/126*sqrt(-x^2 + 1
)/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1) + 1/42*sqrt(-x^2 + 1)/(x^3 - 3*x^2 + 3*x - 1)
- 1/63*sqrt(-x^2 + 1)/(x^2 - 2*x + 1) + 1/63*sqrt(-x^2 + 1)/(x - 1)

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Fricas [A]  time = 0.207367, size = 250, normalized size = 6.1 \[ \frac{7 \, x^{9} - 72 \, x^{8} + 198 \, x^{7} + 252 \, x^{6} - 945 \, x^{5} + 252 \, x^{4} - 84 \, x^{3} - 504 \, x^{2} + 3 \,{\left (3 \, x^{8} - 3 \, x^{7} - 63 \, x^{6} + 203 \, x^{5} - 140 \, x^{3} + 168 \, x^{2} - 336 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} + 1008 \, x}{63 \,{\left (x^{9} - 9 \, x^{8} + 18 \, x^{7} + 18 \, x^{6} - 99 \, x^{5} + 99 \, x^{4} + 24 \, x^{3} - 108 \, x^{2} +{\left (x^{8} - 22 \, x^{6} + 60 \, x^{5} - 39 \, x^{4} - 60 \, x^{3} + 116 \, x^{2} - 72 \, x + 16\right )} \sqrt{x + 1} \sqrt{-x + 1} + 72 \, x - 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/63*(7*x^9 - 72*x^8 + 198*x^7 + 252*x^6 - 945*x^5 + 252*x^4 - 84*x^3 - 504*x^2
+ 3*(3*x^8 - 3*x^7 - 63*x^6 + 203*x^5 - 140*x^3 + 168*x^2 - 336*x)*sqrt(x + 1)*s
qrt(-x + 1) + 1008*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)**(5/2)/(1-x)**(11/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.217918, size = 30, normalized size = 0.73 \[ \frac{{\left (x + 1\right )}^{\frac{7}{2}}{\left (x - 8\right )} \sqrt{-x + 1}}{63 \,{\left (x - 1\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/63*(x + 1)^(7/2)*(x - 8)*sqrt(-x + 1)/(x - 1)^5