3.1124 \(\int \frac{1}{(1-x)^{9/2} (1+x)^{3/2}} \, dx\)

Optimal. Leaf size=82 \[ \frac{8 x}{35 \sqrt{1-x} \sqrt{x+1}}+\frac{4}{35 (1-x)^{3/2} \sqrt{x+1}}+\frac{4}{35 (1-x)^{5/2} \sqrt{x+1}}+\frac{1}{7 (1-x)^{7/2} \sqrt{x+1}} \]

[Out]

1/(7*(1 - x)^(7/2)*Sqrt[1 + x]) + 4/(35*(1 - x)^(5/2)*Sqrt[1 + x]) + 4/(35*(1 -
x)^(3/2)*Sqrt[1 + x]) + (8*x)/(35*Sqrt[1 - x]*Sqrt[1 + x])

_______________________________________________________________________________________

Rubi [A]  time = 0.0508251, antiderivative size = 82, 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 \[ \frac{8 x}{35 \sqrt{1-x} \sqrt{x+1}}+\frac{4}{35 (1-x)^{3/2} \sqrt{x+1}}+\frac{4}{35 (1-x)^{5/2} \sqrt{x+1}}+\frac{1}{7 (1-x)^{7/2} \sqrt{x+1}} \]

Antiderivative was successfully verified.

[In]  Int[1/((1 - x)^(9/2)*(1 + x)^(3/2)),x]

[Out]

1/(7*(1 - x)^(7/2)*Sqrt[1 + x]) + 4/(35*(1 - x)^(5/2)*Sqrt[1 + x]) + 4/(35*(1 -
x)^(3/2)*Sqrt[1 + x]) + (8*x)/(35*Sqrt[1 - x]*Sqrt[1 + x])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 6.5305, size = 68, normalized size = 0.83 \[ \frac{8 x}{35 \sqrt{- x + 1} \sqrt{x + 1}} + \frac{4}{35 \left (- x + 1\right )^{\frac{3}{2}} \sqrt{x + 1}} + \frac{4}{35 \left (- x + 1\right )^{\frac{5}{2}} \sqrt{x + 1}} + \frac{1}{7 \left (- x + 1\right )^{\frac{7}{2}} \sqrt{x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(1-x)**(9/2)/(1+x)**(3/2),x)

[Out]

8*x/(35*sqrt(-x + 1)*sqrt(x + 1)) + 4/(35*(-x + 1)**(3/2)*sqrt(x + 1)) + 4/(35*(
-x + 1)**(5/2)*sqrt(x + 1)) + 1/(7*(-x + 1)**(7/2)*sqrt(x + 1))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0292327, size = 40, normalized size = 0.49 \[ -\frac{8 x^4-24 x^3+20 x^2+4 x-13}{35 (1-x)^{7/2} \sqrt{x+1}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((1 - x)^(9/2)*(1 + x)^(3/2)),x]

[Out]

-(-13 + 4*x + 20*x^2 - 24*x^3 + 8*x^4)/(35*(1 - x)^(7/2)*Sqrt[1 + x])

_______________________________________________________________________________________

Maple [A]  time = 0.005, size = 35, normalized size = 0.4 \[ -{\frac{8\,{x}^{4}-24\,{x}^{3}+20\,{x}^{2}+4\,x-13}{35} \left ( 1-x \right ) ^{-{\frac{7}{2}}}{\frac{1}{\sqrt{1+x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1-x)^(9/2)/(1+x)^(3/2),x)

[Out]

-1/35*(8*x^4-24*x^3+20*x^2+4*x-13)/(1+x)^(1/2)/(1-x)^(7/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.32647, size = 181, normalized size = 2.21 \[ \frac{8 \, x}{35 \, \sqrt{-x^{2} + 1}} - \frac{1}{7 \,{\left (\sqrt{-x^{2} + 1} x^{3} - 3 \, \sqrt{-x^{2} + 1} x^{2} + 3 \, \sqrt{-x^{2} + 1} x - \sqrt{-x^{2} + 1}\right )}} + \frac{4}{35 \,{\left (\sqrt{-x^{2} + 1} x^{2} - 2 \, \sqrt{-x^{2} + 1} x + \sqrt{-x^{2} + 1}\right )}} - \frac{4}{35 \,{\left (\sqrt{-x^{2} + 1} x - \sqrt{-x^{2} + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x + 1)^(3/2)*(-x + 1)^(9/2)),x, algorithm="maxima")

[Out]

8/35*x/sqrt(-x^2 + 1) - 1/7/(sqrt(-x^2 + 1)*x^3 - 3*sqrt(-x^2 + 1)*x^2 + 3*sqrt(
-x^2 + 1)*x - sqrt(-x^2 + 1)) + 4/35/(sqrt(-x^2 + 1)*x^2 - 2*sqrt(-x^2 + 1)*x +
sqrt(-x^2 + 1)) - 4/35/(sqrt(-x^2 + 1)*x - sqrt(-x^2 + 1))

_______________________________________________________________________________________

Fricas [A]  time = 0.207004, size = 219, normalized size = 2.67 \[ -\frac{8 \, x^{8} - 76 \, x^{7} + 112 \, x^{6} + 196 \, x^{5} - 525 \, x^{4} + 140 \, x^{3} + 420 \, x^{2} +{\left (13 \, x^{7} - 7 \, x^{6} - 161 \, x^{5} + 315 \, x^{4} - 420 \, x^{2} + 280 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} - 280 \, x}{35 \,{\left (4 \, x^{7} - 12 \, x^{6} + 32 \, x^{4} - 28 \, x^{3} - 12 \, x^{2} -{\left (x^{7} - 3 \, x^{6} - 5 \, x^{5} + 23 \, x^{4} - 16 \, x^{3} - 16 \, x^{2} + 24 \, x - 8\right )} \sqrt{x + 1} \sqrt{-x + 1} + 24 \, x - 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x + 1)^(3/2)*(-x + 1)^(9/2)),x, algorithm="fricas")

[Out]

-1/35*(8*x^8 - 76*x^7 + 112*x^6 + 196*x^5 - 525*x^4 + 140*x^3 + 420*x^2 + (13*x^
7 - 7*x^6 - 161*x^5 + 315*x^4 - 420*x^2 + 280*x)*sqrt(x + 1)*sqrt(-x + 1) - 280*
x)/(4*x^7 - 12*x^6 + 32*x^4 - 28*x^3 - 12*x^2 - (x^7 - 3*x^6 - 5*x^5 + 23*x^4 -
16*x^3 - 16*x^2 + 24*x - 8)*sqrt(x + 1)*sqrt(-x + 1) + 24*x - 8)

_______________________________________________________________________________________

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)**(9/2)/(1+x)**(3/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.208289, size = 107, normalized size = 1.3 \[ \frac{\sqrt{2} - \sqrt{-x + 1}}{32 \, \sqrt{x + 1}} - \frac{\sqrt{x + 1}}{32 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}} - \frac{{\left ({\left ({\left (93 \, x - 523\right )}{\left (x + 1\right )} + 1400\right )}{\left (x + 1\right )} - 1120\right )} \sqrt{x + 1} \sqrt{-x + 1}}{560 \,{\left (x - 1\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x + 1)^(3/2)*(-x + 1)^(9/2)),x, algorithm="giac")

[Out]

1/32*(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) - 1/32*sqrt(x + 1)/(sqrt(2) - sqrt(-x
+ 1)) - 1/560*(((93*x - 523)*(x + 1) + 1400)*(x + 1) - 1120)*sqrt(x + 1)*sqrt(-x
+ 1)/(x - 1)^4