3.1125 \(\int \frac{1}{(1-x)^{11/2} (1+x)^{3/2}} \, dx\)

Optimal. Leaf size=102 \[ \frac{8 x}{63 \sqrt{1-x} \sqrt{x+1}}+\frac{4}{63 (1-x)^{3/2} \sqrt{x+1}}+\frac{4}{63 (1-x)^{5/2} \sqrt{x+1}}+\frac{5}{63 (1-x)^{7/2} \sqrt{x+1}}+\frac{1}{9 (1-x)^{9/2} \sqrt{x+1}} \]

[Out]

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

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Rubi [A]  time = 0.0680738, antiderivative size = 102, 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 \[ \frac{8 x}{63 \sqrt{1-x} \sqrt{x+1}}+\frac{4}{63 (1-x)^{3/2} \sqrt{x+1}}+\frac{4}{63 (1-x)^{5/2} \sqrt{x+1}}+\frac{5}{63 (1-x)^{7/2} \sqrt{x+1}}+\frac{1}{9 (1-x)^{9/2} \sqrt{x+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.0677, size = 85, normalized size = 0.83 \[ \frac{8 x}{63 \sqrt{- x + 1} \sqrt{x + 1}} + \frac{4}{63 \left (- x + 1\right )^{\frac{3}{2}} \sqrt{x + 1}} + \frac{4}{63 \left (- x + 1\right )^{\frac{5}{2}} \sqrt{x + 1}} + \frac{5}{63 \left (- x + 1\right )^{\frac{7}{2}} \sqrt{x + 1}} + \frac{1}{9 \left (- x + 1\right )^{\frac{9}{2}} \sqrt{x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0359373, size = 45, normalized size = 0.44 \[ \frac{8 x^5-32 x^4+44 x^3-16 x^2-17 x+20}{63 (1-x)^{9/2} \sqrt{x+1}} \]

Antiderivative was successfully verified.

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

[Out]

(20 - 17*x - 16*x^2 + 44*x^3 - 32*x^4 + 8*x^5)/(63*(1 - x)^(9/2)*Sqrt[1 + x])

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Maple [A]  time = 0.004, size = 40, normalized size = 0.4 \[{\frac{8\,{x}^{5}-32\,{x}^{4}+44\,{x}^{3}-16\,{x}^{2}-17\,x+20}{63} \left ( 1-x \right ) ^{-{\frac{9}{2}}}{\frac{1}{\sqrt{1+x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/63*(8*x^5-32*x^4+44*x^3-16*x^2-17*x+20)/(1+x)^(1/2)/(1-x)^(9/2)

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Maxima [A]  time = 1.34834, size = 271, normalized size = 2.66 \[ \frac{8 \, x}{63 \, \sqrt{-x^{2} + 1}} + \frac{1}{9 \,{\left (\sqrt{-x^{2} + 1} x^{4} - 4 \, \sqrt{-x^{2} + 1} x^{3} + 6 \, \sqrt{-x^{2} + 1} x^{2} - 4 \, \sqrt{-x^{2} + 1} x + \sqrt{-x^{2} + 1}\right )}} - \frac{5}{63 \,{\left (\sqrt{-x^{2} + 1} x^{3} - 3 \, \sqrt{-x^{2} + 1} x^{2} + 3 \, \sqrt{-x^{2} + 1} x - \sqrt{-x^{2} + 1}\right )}} + \frac{4}{63 \,{\left (\sqrt{-x^{2} + 1} x^{2} - 2 \, \sqrt{-x^{2} + 1} x + \sqrt{-x^{2} + 1}\right )}} - \frac{4}{63 \,{\left (\sqrt{-x^{2} + 1} x - \sqrt{-x^{2} + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.204996, size = 286, normalized size = 2.8 \[ \frac{20 \, x^{10} - 40 \, x^{9} - 300 \, x^{8} + 1020 \, x^{7} - 420 \, x^{6} - 2037 \, x^{5} + 2688 \, x^{4} + 84 \, x^{3} - 2016 \, x^{2} -{\left (8 \, x^{9} - 132 \, x^{8} + 348 \, x^{7} + 168 \, x^{6} - 1617 \, x^{5} + 1680 \, x^{4} + 588 \, x^{3} - 2016 \, x^{2} + 1008 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} + 1008 \, x}{63 \,{\left (x^{10} - 4 \, x^{9} - 7 \, x^{8} + 48 \, x^{7} - 49 \, x^{6} - 60 \, x^{5} + 139 \, x^{4} - 48 \, x^{3} - 68 \, x^{2} +{\left (5 \, x^{8} - 20 \, x^{7} + 10 \, x^{6} + 60 \, x^{5} - 99 \, x^{4} + 16 \, x^{3} + 76 \, x^{2} - 64 \, x + 16\right )} \sqrt{x + 1} \sqrt{-x + 1} + 64 \, x - 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/63*(20*x^10 - 40*x^9 - 300*x^8 + 1020*x^7 - 420*x^6 - 2037*x^5 + 2688*x^4 + 84
*x^3 - 2016*x^2 - (8*x^9 - 132*x^8 + 348*x^7 + 168*x^6 - 1617*x^5 + 1680*x^4 + 5
88*x^3 - 2016*x^2 + 1008*x)*sqrt(x + 1)*sqrt(-x + 1) + 1008*x)/(x^10 - 4*x^9 - 7
*x^8 + 48*x^7 - 49*x^6 - 60*x^5 + 139*x^4 - 48*x^3 - 68*x^2 + (5*x^8 - 20*x^7 +
10*x^6 + 60*x^5 - 99*x^4 + 16*x^3 + 76*x^2 - 64*x + 16)*sqrt(x + 1)*sqrt(-x + 1)
+ 64*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)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.209557, size = 115, normalized size = 1.13 \[ \frac{\sqrt{2} - \sqrt{-x + 1}}{64 \, \sqrt{x + 1}} - \frac{\sqrt{x + 1}}{64 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}} - \frac{{\left ({\left ({\left ({\left (193 \, x - 1481\right )}{\left (x + 1\right )} + 5544\right )}{\left (x + 1\right )} - 8400\right )}{\left (x + 1\right )} + 5040\right )} \sqrt{x + 1} \sqrt{-x + 1}}{2016 \,{\left (x - 1\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/64*(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) - 1/64*sqrt(x + 1)/(sqrt(2) - sqrt(-x
+ 1)) - 1/2016*((((193*x - 1481)*(x + 1) + 5544)*(x + 1) - 8400)*(x + 1) + 5040)
*sqrt(x + 1)*sqrt(-x + 1)/(x - 1)^5